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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 586–607 | Cite as

Random Walk in Balls and an Extension of the Banach Integral in Abstract Spaces

  • Tadeusz Banek
  • August M. ZapałaEmail author
Open Access
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Abstract

We describe the construction of a random walk in a Banach space \(\mathbb {B}\) with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in \(L^{p}\left( \mathbb {B}\right) ,\,1\le p\,<\infty ,\) to a random element \(\xi ,\) which generates a probability measure with support contained in the unit ball \(B\subset \mathbb {B}\). Moreover, we define the Banach integral with respect to the distribution of \(\xi \) for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces \(\ell ^{p},\,1\le p < \infty \) and \(L^{p}[0,1],\,1< p < \infty \) are given.

Keywords

Banach random walk Martingale Radon–Nikodym property Quasi-orthogonal Schauder basis 

Mathematics Subject Classification (2010)

Primary 60J15 60B12 60G42 Secondary 60G46 

1 Introduction

Let \(\left\{ e_{i},i\ge 1\right\} \) be a complete orthonormal system (CONS) in a separable Hilbert space \( \left( \mathbb {H},\left| \cdot \right| \right) \) and let \(\pi _{n}\left( x\right) =\sum _{i=1}^{n}x_{i}e_{i}\) for \(x=\sum _{i=1}^{\infty }x_{i}e_{i}\in \mathbb {H}\). Denote by \(K_{n}\) and B the unit balls in \( \mathbb {R}^{n}\) and \(\mathbb {H}\), respectively.

In a paper published as addendum to the Saks monograph Theory of the Integral, Banach [1] described the most general form of a nonnegative linear functional F (satisfying additional conditions analogous to some properties of the Lebesgue integral, thus called by Banach \(\mathfrak {L}\)-integral), defined on the linear set \(\mathcal {L}\) of bounded, Borel measurable functions \(\varPhi {:}\,B\rightarrow \mathbb {R}\), namely
$$\begin{aligned} F\left( \varPhi \right) =\lim _{n\rightarrow \infty }F_{n}\left( \varPhi \right) , \end{aligned}$$
where
$$\begin{aligned}&F_{n}\left( \varPhi \right) =\int _{K_{n}}\varPhi \left( \pi _{n}\left( x\right) \right) g_{n}\left( x_{1},\ldots ,x_{n}\right) \mathrm{d}x_{1}\ldots \mathrm{d}x_{n},\\&g_{n}\left( x_{1},\ldots ,x_{n}\right) \\&\quad =\mathbb {1}_{K_{n}}\left( x_{1},\ldots ,x_{n}\right) \frac{g\left( x_{1}\right) g\left( x_{2}{/}\sqrt{1-x_{1}^{2}}\right) \ldots \, g\left( x_{n}{/}\sqrt{ 1-\left( x_{1}^{2}+\cdots +x_{n-1}^{2}\right) }\right) }{\sqrt{\left( 1-x_{1}^{2}\right) \cdot \left[ 1-\left( x_{1}^{2}+x_{2}^{2}\right) \right] \cdot \ldots \cdot \left[ 1-\left( x_{1}^{2}+\cdots +x_{n-1}^{2}\right) \right] }}, \end{aligned}$$
\(g{:}\,\left[ -1,1\right] \rightarrow \left[ 0,\infty \right) \) is Borel measurable and integrable with \(\int _{-1}^{1}g\left( t\right) \mathrm{d}t=1,\) and \( \mathbb {1}_{A}\) denotes the indicator of the set A.
In fact, Banach [1] considered only the case when g is the density of the uniform distribution on \([-\,1,1]\), and a more general case was treated by Banek [2]. Furthermore, Banek [2] observed that
$$\begin{aligned} F_{n}\left( \varPhi \right) =E\varPhi \left( Z_{n}\right) , \end{aligned}$$
where \(\left\{ Z_{n} \right\} \) is the so-called Banach random walk (BRW) in \(B\subset \mathbb {H}\) given by the random linear combination \( Z_{n}=\sum _{i=1}^{n}X_{i}e_{i},~ n\ge 1, \) of elements of CONS \(\{e_{i} \}\) in \(\mathbb {H}\), with coefficients \( X_{i}\) that are dependent r.v.’s defined recursively as follows: \(X_{1}\) is a r.v. having density g concentrated on the interval \(\left[ -\,1,1\right] \), and if the r.v.’s \(X_{1},\ldots ,X_{n-1}\) are already defined, then \(X_{n}\) is defined as a r.v. with probability density \(g\left( x_{n}{/}\sqrt{1-(X_{1}^{2}+\cdots +X_{n-1}^{2})}\right) \) in the random interval
$$\begin{aligned} \left[ -\sqrt{ 1-\left( X_{1}^{2}+\cdots +X_{n-1}^{2}\right) },\sqrt{1-\left( X_{1}^{2}+\cdots +X_{n-1}^{2}\right) }\right] . \end{aligned}$$
The last observation forms a probabilistic background to the purely deterministic Banach construction of the \( \mathfrak {L}\)-integral for a class of bounded, Borel measurable functions defined in \(B\subset \mathbb {H}\).

It is worth mentioning that Banach [1] considered two various special cases: (1) the mapping \(\varPhi \) is defined on a compact metric space and (2) \(\varPhi \) is defined in the unit ball of a separable Hilbert space.

In this paper, we describe a generalized BRW \(\left\{ Z_{n},n\ge 1\right\} \) with values in the unit ball of a Banach space. Moreover, we give a criterion for the existence of the Banach–Lebesgue integral
$$\begin{aligned} F\left( \varPhi \right) =\lim _{n\rightarrow \infty }F_{n}\left( \varPhi \right) \end{aligned}$$
(1)
in terms of the constructed BRW \(\left\{ Z_{n},n\ge 1\right\} \), where \( \varPhi \) is a bounded, Borel measurable real-valued function defined in the unit ball of a Banach space.

2 Banach Random Walk in a Banach space

Let \(\left( \mathbb {B},\left\| \cdot \right\| \right) \) be an infinite-dimensional Banach space with a Schauder basis \(\left\{ b_{n},n\ge 1\right\} \). Then each vector \(x\in \mathbb {B}\) possesses a unique series expansion \(x=\sum _{k=1}^{\infty }x_{k}b_{k},\) and thus, for all \(n\ge 1\), the projections \(\pi _{n}{:}\,\mathbb {B}\rightarrow \mathbb {B},\) given by \(\pi _{n}\left( x\right) =\sum _{k=1}^{n}x_{k}b_{k}\), are well defined. Denote
$$\begin{aligned} B_{n}=\left\{ \pi _{n}\left( x\right) \in \mathbb {B}{:}\,\left\| \pi _{n}\left( x\right) \right\| \le 1\right\} ,\quad B=\left\{ x\in \mathbb { B}{:}\,\left\| x\right\| \le 1\right\} , \end{aligned}$$
and put
$$\begin{aligned} \alpha _{1}=\inf \left\{ t\in \mathbb {R}{:}\,\left\| tb_{1}\right\| \le 1\right\} ,\quad \beta _{1}=\sup \left\{ t\in \mathbb {R}{:}\,\left\| tb_{1}\right\| \le 1\right\} \left( =-\alpha _{1}\right) . \end{aligned}$$
Furthermore, given any point \(\pi _{n-1}\left( x\right) \in B_{n-1},\,n\ge 2\), define inductively
$$\begin{aligned} \alpha _{n}= & {} \alpha _{n}(\pi _{n-1}\left( x\right) ) =\, \inf \left\{ t\in \mathbb {R}{:}\,\left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1\right\} ,\\ \beta _{n}= & {} \beta _{n}(\pi _{n-1}\left( x\right) ) =\, \sup \left\{ t\in \mathbb {R}{:}\,\left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1\right\} . \end{aligned}$$
It is clear that \(\alpha _{n}\le 0\le \beta _{n}\), and \(\left[ \alpha _{n},\beta _{n}\right] ,\,n\ge 1\), are bounded intervals in \(\mathbb {R}\), for
$$\begin{aligned} \left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1 \,\, \Rightarrow \,\, \left| t\right| \cdot \left\| b_{n}\right\| \le \left\| \pi _{n-1}\left( x\right) \right\| +1, ~ n\ge 1, \end{aligned}$$
where \(\pi _{n-1}\left( x\right) = \pi _{0}\left( x\right) = 0\) for \(n=1\). Obviously, \(\alpha _{n}\) and \(\beta _{n}\) depend on \(\pi _{n-1}\left( x\right) \in B_{n-1}\) and \(b_n\), and in general the intervals \(\left[ \alpha _{n},\beta _{n}\right] ,\,n\ge 2\), need not be symmetric about 0. In addition, it may happen that for some \(n\ge 2\) the interval \(\left[ \alpha _{n},\beta _{n}\right] \) reduces to the single point \(\left[ 0,0\right] =\left\{ 0\right\} \). To fix a standard length of the first interval, without loss of generality we may and do assume that \(\left\| b_{1}\right\| =1\) (but we do not require that \(\left\| b_{n}\right\| =1\) for all \(n\ge 2\)). In such a situation, \(\alpha _{1}=-1\) and \(\beta _{1}=1\).
Let \(G_{n},\,n\ge 1\), be arbitrary probability distributions concentrated on \(\left[ -\,1,1\right] \subset \mathbb {R}\), i.e., \(G_{n}\left( \left[ -\,1,1\right] \right) =1\) for all \(n\ge 1\). Define inductively on a probability space \(\left( \varOmega , \mathcal {F},P\right) \) a sequence of (dependent) real r.v.’s \(\left\{ X_{n},n\ge 1\right\} \) and, associated with it, a sequence of \(\mathbb {B}\)-valued random elements (r.e.) \(\left\{ Z_{n},n\ge 1\right\} \). Namely, let \(X_{1}\) be a r.v. with distribution \(G_{1}\) and let \(Z_{1}=X_{1}b_{1}\). Then \(X_{1}\left( \omega \right) \in \left[ \alpha _{1},\beta _{1}\right] =\left[ -1,1\right] \) a.s., and thus, we may define \(X_{2}\) as a r.v. distributed according to \(G_{2},\) scaled linearly in such a way that it is concentrated on \(\left[ \alpha _{2},\beta _{2} \right] =\left[ \alpha _{2}\left( Z_{1}\left( \omega \right) \right) ,\beta _{2}\left( Z_{1}\left( \omega \right) \right) \right] \). In other words, \(X_{2}\) is a r.v. with distribution function
$$\begin{aligned} G_{2}\left( \frac{2t- \left[ \beta _{2}\left( Z_{1}\left( \omega \right) \right) +\alpha _{2}\left( Z_{1}\left( \omega \right) \right) \right] }{\beta _{2}\left( Z_{1}\left( \omega \right) \right) -\alpha _{2}\left( Z_{1}\left( \omega \right) \right) } \right) ,~ t\in \mathbb {R}, \end{aligned}$$
whenever \(\beta _2 - \alpha _2 > 0\), and then we put \(Z_{2}=X_{1}b_{1}+X_{2}b_{2}\). Next, given any value \(X_{2}\left( \omega \right) \), and a fortiori\(Z_{2}\left( \omega \right) \in B_{2}\) a.s., we define \(X_{3}\) as a r.v. with distribution \(G_{3}\) scaled linearly in such a way that it is concentrated on the interval\(\left[ \alpha _{3},\beta _{3}\right] =\left[ \alpha _{3}\left( Z_{2}\left( \omega \right) \right) ,\beta _{3}\left( Z_{2}\left( \omega \right) \right) \right] ,\) and then put \(Z_{3}=X_{1}b_{1}+X_{2}b_{2}+X_{3}b_{3},\) etc. More generally, if \(X_{1}, \ldots ,X_{n-1}\) and \(Z_{1}, \ldots ,Z_{n-1}\) are already defined in such a manner that \(Z_{n-1}\left( \omega \right) \in B_{n-1}\) a.s., then \( X_{n}\) is a r.v. with distribution function
$$\begin{aligned} G_{n}\left( \frac{2t- \left[ \beta _{n}\left( Z_{n-1}\left( \omega \right) \right) +\alpha _{n}\left( Z_{n-1}\left( \omega \right) \right) \right] }{\beta _{n}\left( Z_{n-1}\left( \omega \right) \right) -\alpha _{n}\left( Z_{n-1}\left( \omega \right) \right) } \right) , ~ t\in \mathbb {R}, \end{aligned}$$
provided \(\beta _n - \alpha _n > 0\) and \(Z_{n}=X_{1}b_{1}+X_{2}b_{2}+\cdots +X_{n}b_{n} = Z_{n-1} + X_{n}b_{n}.\)

As was already mentioned, it may happen that for some \(n\ge 1\) and \(Z_{n}\left( \omega \right) \in B_{n}\) the interval \(\left[ \alpha _{n+1},\beta _{n+1}\right] = \left[ \alpha _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right] \) is reduced to the single point \(\left\{ 0\right\} \); in this situation, we assume that the distribution \( G_{n+1}\) is transformed in such a way that it assigns the unit mass to the one point set \(\{0\}\). Although in such a case \(Z_{n+1}(\omega ) = Z_n (\omega )\), the next random interval \([\alpha _{n+2},\beta _{n+2} ] = [\alpha _{n+2}\left( Z_{n+1}(\omega )\right) ,\beta _{n+2}\left( Z_{n+1}(\omega )\right) ]\), defined by means of the successive basic vector \(b_{n+2}\), need not be equal to \(\{0\}\); thus, the process is still continued.

Definition 1

The sequence of \(\mathbb {B}\)-valued r.e.’s \(\left\{ Z_{n},n\ge 1\right\} \) obtained in the way described above is called Banach random walk (BRW) in a Banach space \(\mathbb {B}\).

It seems that the main idea of Banach’s [1] construction of \(\mathfrak {L}\)-integral was the symmetry of mappings corresponding to the symmetry of Lebesgue measures in \(\mathbb {R}^{n},\,n\ge 1\), which led to convergence of the integral functional in (1). Therefore, we introduce in addition the following notion:

Definition 2

The Schauder basis \( \left\{ b_{n},n\ge 1\right\} \) is called quasi-orthogonal, if
$$\begin{aligned} \beta _{n+1}(\pi _{n}\left( x\right) )= -\alpha _{n+1}\left( \pi _{n}\left( x\right) \right) \end{aligned}$$
(2)
for all \(n\ge 1\) and \(x\in \mathbb {B}\) such that \(\pi _{n}\left( x\right) \in B_n\).
Recall that the basis \(\left\{ b_{n},n\ge 1\right\} \) in a Banach space \((\mathbb {B}, \Vert \cdot \Vert )\) is said to be unconditional, if for all \(n\ge 1,\,x_{k}\in \mathbb {R}\) and \(\epsilon _{k}=\pm 1,\,1\le k\le n,\) we have
$$\begin{aligned} \left\| \sum \limits _{k=1}^{n}\epsilon _{k}x_{k}b_{k}\right\| =\left\| \sum \limits _{k=1}^{n}x_{k}b_{k}\right\| . \end{aligned}$$
(3)
The definition of a quasi-orthogonal Schauder basis in a Banach space seems to be similar to the condition defining an unconditional basis, but in spite of this, these two notions are not equivalent. If the basis \(\left\{ b_{n},n\ge 1\right\} \) is unconditional, i.e., (3) holds, then it is obviously quasi-orthogonal, but the converse need not be true. To explain the notion of quasi-orthogonality, below we present the construction of relevant examples of Banach spaces with quasi-orthogonal Schauder bases which are not unconditional. It should be pointed out that many familiar Banach spaces possess unconditional Schauder bases (thus in fact quasi-orthogonal) consisting of unit vectors, but the class of Banach spaces with quasi-orthogonal bases is substantially larger than the class of spaces with unconditional bases.

3 Banach Spaces with Quasi-Orthogonal Bases

The quasi-orthogonal basis is constructed sequentially, step by step: given any basic vectors \(b_{1},b_{2},\ldots ,b_{n},\) the element \(b_{n+1}\) of the basis is chosen in such a way that for arbitrary \(x_{1},x_{2},\ldots ,x_{n} \in \mathbb {R}\) satisfying condition \( x_{1}b_{1}+ \cdots +x_{n}b_{n}=\pi _n (x)\in B_n,\) Eq. (2) is satisfied. As will be seen condition (3) for this property is not necessary.

1. Spaces of bounded sums and conditionally convergent series

Let \(S=\mathbb {R}^{\mathbb {N}}=\left\{ x=\left( x_{1},x_{2},\ldots \right) {:}\,x_{k}\in \mathbb {R}\text { for all }k\ge 1\right\} \ \)be the set of all infinite sequences of real numbers. Define a function \(\left\| \cdot \right\| {:}\,S\rightarrow \left[ 0,\infty \right] \) by
$$\begin{aligned} \left\| x\right\|= & {} \sup \left\{ \left| x_{1}+x_{2}\right| ,\left| x_{1}-x_{2}\right| ,\left| x_{1}+x_{2}+x_{3}\right| ,\left| x_{1}+x_{2}-x_{3}\right| ,\right. \\&\qquad \left. \ldots ,\left| x_{1}+\cdots +x_{n-1}+x_{n}\right| ,\left| x_{1}+\cdots +x_{n-1}-x_{n}\right| ,\ldots \right\} \\= & {} \sup \left\{ \left| x_{1}\right| + \left| x_{2}\right| , \left| x_{1}+x_{2}\right| + \left| x_{3}\right| , \ldots , \left| x_{1}+\cdots +x_{n}\right| + \left| x_{n+1} \right| , \ldots \right\} , \end{aligned}$$
and next put
$$\begin{aligned} S_{b}=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S{:}\,\left\| x\right\| <\infty \right\} . \end{aligned}$$
Then \(\left\| \cdot \right\| \) is a norm in \(S_{b}\), and \(\left( S_{b},\left\| \cdot \right\| \right) \) is a Banach space. The space \( S_{b}\) consists of all bounded sequences \(\left( x_{1},x_{2},\ldots \right) \in S\) of real numbers with bounded partial sums \(s_{n}=x_{1}+\cdots +x_{n};\) namely, if \(x\in S_{b}\) and \(\left\| x\right\| =M<\infty \), then \(\left| x_{n}\right| \le M\) and \(\left| s_{n}\right| \le M\) for all \( n\ge 1\). Conversely, if there exists a constant \(0\le M<\infty \) such that \(\left| x_{n}\right| \le M\) and \(\left| s_{n}\right| \le M\) for all \(n\ge 1,\) then \(\left| s_{n}\pm x_{n+1}\right| \le \left| s_{n}\right| +\left| x_{n+1}\right| \le 2M,\) and thus, \( \left\| x\right\| \le 2M,\) i.e., \(x=\left( x_{1},x_{2},\ldots \right) \in S_{b}.\) Therefore, \(\left( S_{b},\left\| \cdot \right\| \right) \) may be called the space of bounded sums.

However, the space \(S_{b}\) of sequences of real numbers is not separable. To show this, consider the family \(2^{\mathbb {N}}\) of all the subsets of the set \(\mathbb {N}=\left\{ 1,2,\ldots \right\} ,\) and for \(\emptyset \ne A=\left\{ k_{1},k_{2},\ldots \right\} \subseteq \mathbb {N}\), where \(k_1< k_2 < \cdots \), define \(x_{A}\) as the sequence with terms \(x_{k_{2j-1}}=1,\,x_{k_{2j}}=-1,\,j\ge 1,\) and \(x_{i}=0\) otherwise, along with \(x_{\emptyset }=(0,0,\ldots ).\) Then \(\left\| x_{A}\right\| \le 2\), while \(\left\| x_{A}-x_{B}\right\| \ge 1\) whenever \(A,B\subseteq \mathbb {N},\,A\ne B.\) Since the set \(2^{\mathbb {N}}\) is uncountable, the space \(S_{b}\) is nonseparable. Hence in the context of our requirements, the space \(S_{b}\) is inadequate.

Consider the set
$$\begin{aligned} S_{c}=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S_{b}{:}\,\text { the series } \sum \limits _{k}x_{k}\text { converges }\right\} . \end{aligned}$$
In other words, \(S_{c}\) is the set of all sequences \(x=\left( x_{1},x_{2},\ldots \right) \in S_{b}\) for which a finite limit \( \lim _{n}s_{n}=s\in \mathbb {R}\) exists. It can be easily verified that \( \left( S_{c},\left\| \cdot \right\| \right) \) is also a Banach space. Moreover, the space \(S_{c}\) is separable. Indeed, the set of elements
$$\begin{aligned} \left\{ e_{n}=\left( \delta _{kn},k\ge 1\right) ,\,n=1,2,\ldots \right\} , \end{aligned}$$
where \(\delta _{kn}=0\) for \(k\ne n\) and \(\delta _{nn}=1,\) is a basis of the space \(S_{c},\) and finite linear combinations of vectors \(e_{n}\) with rational coefficients from a countable dense subset in \(S_{c}.\) Moreover, from the definition of the norm \(\left\| \cdot \right\| \) it follows that the basis \(\left\{ e_{n},\,n\ge 1\right\} \) is quasi-orthogonal, but it is not unconditional, because sums of the form \(\epsilon _{1}x_{1}+\epsilon _{2}x_{2}+\cdots +\epsilon _{n}x_{n},\,x=\left( x_{1},x_{2},\ldots \right) \in S_{c},\) need not be convergent for all combinations of signs \(\epsilon _{k}=\pm 1\). Taking into account the above properties, \(\left( S_{c},\left\| \cdot \right\| \right) \) may be called the Banach space of conditionally convergent series.
Since the existence of \(\lim _{n}s_{n}=s\in \mathbb {R}\) implies that \( \lim _{n}x_{n}=0\), we conclude that \(x\in S_{c}\Rightarrow x\in c_{0}.\) The Banach space \(c_{0}\) of sequences of real numbers convergent to 0 is usually considered with the supremum norm \(\left| x\right| _{\infty }=\sup \left\{ \left| x_{1}\right| ,\left| x_{2}\right| ,\ldots \right\} ,\) but the two norms \(\left\| \cdot \right\| \) and \( \left| \cdot \right| _{\infty }\) restricted to \(S_{c}\) are not equivalent. To see this, consider the sequence of points \(\left\{ x^{\left( n\right) },n\ge 1\right\} ,\)
$$\begin{aligned} x^{\left( 1\right) }= & {} \left( 1,-1/2,1/3,-1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\ x^{\left( 2\right) }= & {} \left( 1,1/2,1/3,-1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\ x^{\left( 3\right) }= & {} \left( 1,1/2,1/3,1/4,1/5,-1/6,1/7,-1/8,\ldots \right) , \\&\qquad \vdots \end{aligned}$$
and put \(x^{\left( \infty \right) }=\left( 1,1/2,1/3,1/4,1/5,1/6,\ldots ,1/n,1/\left( n+1\right) ,\ldots \right) .\)

Then \(\left\| x^{\left( n\right) }\right\| <\infty \) for all \(n\ge 1,\) so that \(\left\{ x^{\left( n\right) },n\ge 1\right\} \subset S_{c}.\) Moreover, \(\left| x^{\left( n\right) }-x^{\left( \infty \right) }\right| _{\infty }=1/n\rightarrow 0,\) while \(\left\| x^{\left( n\right) }-x^{\left( \infty \right) }\right\| =\infty ,\,n=1,2,\ldots \) which is a consequence of the fact that \(\sum \nolimits _{n}1/n=\infty .\) Therefore, the inclusion \(S_{c}\subset c_{0}\) is valid only for sets, but it is not true for Banach spaces, \(\left( S_{c},\left\| \cdot \right\| \right) \varsubsetneq \left( c_{0},\left| \cdot \right| _{\infty }\right) .\)

A similar effect as in the case of the space \(S_{c}\) can be obtained for every fixed system of signs \(\epsilon =\left( \epsilon _{1},\epsilon _{2},\ldots \right) \in \left\{ -1,1\right\} ^{\mathbb {N}}\) and the norm given by
$$\begin{aligned} \left\| x\right\| _{\epsilon }=\left\| \left( \epsilon _{1}x_{1},\epsilon _{2}x_{2},\epsilon _{3}x_{3},\ldots \right) \right\| ~ \text {for} ~ x=(x_1,x_2, \ldots ). \end{aligned}$$
In this way, we obtain a Banach space \(\left( S_{c,\epsilon },\left\| \cdot \right\| _{\epsilon }\right) ,\) where
$$\begin{aligned} S_{c,\epsilon }=\left\{ x=\left( x_{1},x_{2},\ldots \right) \in S{:}\,\left\| x\right\| _{\epsilon }<\infty \text { and the series }\sum \limits _{k} \epsilon _{k}x_{k}\text { converges}\right\} . \end{aligned}$$
The basis \(\left\{ e_{n},\,n\ge 1\right\} \) in \(S_{c,\epsilon }\) is quasi-orthogonal, but it is not unconditional. Note now that \(\ell ^{1}\subseteq S_{c,\epsilon }\) for each \(\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}\), therefore \( \ell ^{1}\subseteq \bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{ \mathbb {N}}}S_{c,\epsilon }. \) On the other hand, assuming that \(x=\left( x_{1},x_{2},\ldots \right) \in \displaystyle \bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }\) and taking \(\epsilon ^{(x)}=\left( \mathrm {sign\,}x_{1}, \mathrm {sign\,}x_{2},\ldots ,\mathrm {sign\,}x_{n},\ldots \right) \) we have
$$\begin{aligned} \epsilon _{1}^{(x)}x_{1}+\epsilon _{2}^{(x)}x_{2}+\cdots +\epsilon _{n}^{(x)}x_{n}=\left| x_{1}\right| +\left| x_{2}\right| +\cdots +\left| x_{n}\right| , ~ n\ge 1. \end{aligned}$$
Hence, we infer that
$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left| x_{n}\right| =\sup \limits _{n}\left\{ \left| \epsilon _{1}^{(x)}x_{1}+\epsilon _{2}^{(x)}x_{2}+\cdots +\epsilon _{n}^{(x)}x_{n}\right| \right\} \le \left\| x\right\| _{\epsilon ^{(x)}}, \end{aligned}$$
where \(\epsilon ^{(x)}\in \left\{ -1,1\right\} ^{\mathbb {N}}\), thus \(x\in \ell ^{1}.\) Consequently, \( \ell ^{1}=\bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }. \)
Define next a function \(\left\| \cdot \right\| _{\cap }{:}\,\bigcap \nolimits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }\rightarrow \left[ 0,\infty \right] \) by the formula: \(\left\| x\right\| _{\cap }=\sup \left\{ \left\| x\right\| _{\epsilon },\epsilon \in \left\{ -1,1\right\} ^{ \mathbb {N}}\right\} \). Since
$$\begin{aligned} \left| \epsilon _{1}x_{1}+\epsilon _{2}x_{2}+\cdots +\epsilon _{n-1}x_{n-1}\pm \epsilon _{n}x_{n}\right| \le \left| x_{1}\right| +\left| x_{2}\right| +\cdots +\left| x_{n}\right| \end{aligned}$$
for each \(\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}},\) where the inequality \(\le \) may be replaced by the equality whenever \( \epsilon _{k}=\mathrm {sign\,}x_{k},\,1\le k\le n-1\) and \(\pm \,\epsilon _{n}=\mathrm {sign\,}x_{n}\), the function \(\left\| \cdot \right\| _{\cap }\) assumes only finite values and in fact \(\left\| x\right\| _{\cap }=\sum \nolimits _{n}\left| x_{n}\right| ,\) i.e., \(\left\| \cdot \right\| _{\cap }\) is the norm equal precisely to the norm \( \left| x\right| _{1}=\sum \nolimits _{n}\left| x_{n}\right| \) of the space \(\ell ^{1}.\) Therefore, one can write
$$\begin{aligned} \left( \ell ^{1},\left| \cdot \right| _{1}\right) =\bigcap \limits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}}\left( S_{c,\epsilon },\left\| \cdot \right\| _{\epsilon }\right) . \end{aligned}$$
In this sense, the Banach space \(\left( \ell ^{1},\left| \cdot \right| _{1}\right) \) in comparison with the space \(\left( S_{c},\left\| \cdot \right\| \right) \) is “relatively small”. It is also worth mentioning that the basis \(\left\{ e_{n},n\ge 1\right\} \) in \(S_{c}\) (as well as in \(S_{c,\epsilon }\)) is monotone, i.e., for every choice of scalars \(\{x_{n},n\ge 1\}\subseteq \mathbb {R}\), the sequence of numbers \(\left\{ \Vert \sum \nolimits _{k=1}^{n}x_{n}e_{n}\Vert ,n\ge 1\right\} \) is nondecreasing.

Recall now that a basis \(\{b_{n},n\ge 1\}\) of a Banach space \((\mathbb {B},\Vert \cdot \Vert )\) is called boundedly complete if, for every sequence of scalars \(\{x_{n},n\ge 1\}\subseteq \mathbb {R}\) such that \(\sup _{n\ge 1}\Vert \sum \nolimits _{k=1}^{n}x_{k}b_{k}\Vert <\infty \), the series \( \sum \nolimits _{n=1}^{\infty }x_{n}b_{n}\) converges in norm of \(\mathbb {B}\). Unfortunately, the basis \(\{e_{n},n\ge 1\}\) in \(S_{c}\) or \(S_{c,\epsilon }\) is not boundedly complete.

2. Spaces of conditionally convergent series with rates of convergence

The space \(S_{c}\) described here may be the prototype for a wide class of various spaces with quasi-orthogonal Schauder bases that are not unconditional. For instance, consider the spaces \(S_{c}^{p}\) of (conditionally) summable sequences of real numbers spanned on vectors of the basis \(\left\{ e_{n},n\ge 1\right\} ,\) with norms likewise in \(\ell ^{p},\, 1\le p<\infty ,\) (that describe rates of convergence)
$$\begin{aligned} \left\| x\right\| _{p}=\left( \sum \limits _{~n=1}^{\infty }\left\| R_{n}\left( x\right) \right\| ^{p}\right) ^{1/p}, \end{aligned}$$
where \(R_{n}\left( x\right) =\sum _{k=n}^{\infty }x_{k}e_{k}\) for \(x=\left( x_{1},x_{2},\ldots \right) \in S\); spaces \(S_{c,w}\) with norms determined by some positive weights \(w=\left( w_{1},w_{2},\ldots \right) ,\,w_{i}>0,\)
$$\begin{aligned} \left\| x\right\| _{1,w}=\sum \limits _{n=1}^{\infty }w_{n}\left\| R_{n}\left( x\right) \right\| , \end{aligned}$$
say geometrical weights \(w=\left( w_{1},w_{2},\ldots \right) =\left( q^{1},q^{2},\ldots \right) ,\,q>0,\) or more generally, spaces \(S_{c,w}^{p}\) equipped with norms
$$\begin{aligned} \left\| x\right\| _{p,w}=\left( \sum \limits _{~n=1}^{\infty }w_{n}\left\| R_{n}\left( x\right) \right\| ^{p}\right) ^{1/p}, ~ 1\le p<\infty , ~ \text {etc.} \end{aligned}$$
Since convergence of the series on the right-hand side of the definition of \( \left\| x\right\| _{p}\) implies that partial sums of the series \( \sum \nolimits _{n}x_{n}\) satisfy Cauchy’s criterion, the basis \(\left\{ e_{n},n\ge 1\right\} \) in \(\left( S_{c}^{p},\left\| \cdot \right\| _{p}\right) \) is evidently monotone and boundedly complete. Thus in view of Dunford’s theorem [4, Ch. III, §1, Th. 6, p. 64], these spaces possess the Radon–Nikodym property (RNP)—see [4, Ch. III, §1, p. 61] for the definition of this notion. Moreover, the equality
$$\begin{aligned} \left\| \sum \limits _{j=k}^{n}x_{j}e_{j}\right\| =\left\| \sum \limits _{j=k}^{n-1}x_{j}e_{j}-x_{n}e_{n}\right\| \end{aligned}$$
valid for all \(\left( x_{1},x_{2},\ldots \right) \in S\) and \(1\le k<n<\infty \) implies that the basis \(\left\{ e_{n},n\ge 1\right\} \) in \(\left( S_{c}^{p},\left\| \cdot \right\| _{p}\right) \) is again quasi-orthogonal, but it is not unconditional. A similar conclusion can be also derived for spaces equipped with norms \(\left\| \cdot \right\| _{p,w},\) provided that w is a suitable sequence of weights.

3. Spaces of bounded sums of functions and conditionally convergent function series

Let \(\left\{ q_{1}=0,q_{2}=1,q_{3},q_{4},\ldots \right\} \subset \left[ 0,1 \right] \subset \mathbb {R}\) be a countable set of numbers dense in [0, 1] (arranged in any order), and let \(\left\{ e_{n}\left( t\right) ,n\ge 1\right\} \) be the system of Schauder hat functions in \(\left[ 0,1\right] \) defined as follows: \(e_{1}\left( t\right) =1-t\) and \(e_{2}\left( t\right) =t,\,0\le t\le 1;\) for \(n>2\) the points \(q_{1},\ldots ,q_{n-1}\) divide the interval \(\left[ 0,1\right] \) into \( n-2 \) subintervals, and if \(\left[ q_{i},q_{j}\right] \) is the subinterval which contains the point \(q_{n}\), then \(e_{n}\left( t\right) =0\) for \(t\in [0, q_{i} ] \cup [ q_{j}, 1 ],\,e_{n}\left( q_{n}\right) =1,\) and \(e_{n}\) is a linear function in the interval \(\left[ q_{i},q_{n} \right] \) as well as in \(\left[ q_{n},q_{j}\right] \). It is known that the described system of Schauder hat functions forms a basis for the space \(C\left[ 0,1\right] \) of real-valued continuous functions in \(\left[ 0,1\right] \) with the supremum norm
$$\begin{aligned} \left\| f\right\| _{\infty } =\sup _{0\le t\le 1}\left| f\left( t\right) \right| , \end{aligned}$$
see, e.g., [10, Prop. 2.3.5, p. 29].
Now with every sequence \(x=\left( x_{1},x_{2},\ldots \,\right) \subset \mathbb {R}^{ \mathbb {N}}\), we associate an element g having coefficients \(\left( x_{1},x_{2},\ldots \,\right) \) in the basis \(\left\{ e_{n}\left( t\right) ,n\ge 1\right\} ,\) formally written as \(g:=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right) \), and define
$$\begin{aligned} \left\| g\right\|= & {} \left\| \sup \left\{ \left| x_{1}e_{1}+x_{2}e_{2}\right| ,\left| x_{1}e_{1}-x_{2}e_{2}\right| , \ldots ,\left| x_{1}e_{1}+\cdots +x_{n-1}e_{n-1}+x_{n}e_{n}\right| ,\right. \right. \\&\qquad \left. \left. \left| x_{1}e_{1}+\cdots +x_{n-1}e_{n-1}-x_{n}e_{n}\right| ,\ldots \right\} \right\| _{\infty } \\= & {} \left\| \sup \limits _{n\ge 1} \left\{ \left| x_{1}e_{1}+\cdots +x_{n}e_{n}\right| +\left| x_{n+1}e_{n+1}\right| \right\} \right\| _{\infty }, \end{aligned}$$
along with
$$\begin{aligned} S_{b}\left[ 0,1\right] =\left\{ g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right) {:}\,\left\| g\right\| <\infty \right\} . \end{aligned}$$
Then \(\left( S_{b}\left[ 0,1\right] ,\left\| \cdot \right\| \right) \) is a (nonseparable) Banach space. The space \(S_{b}\left[ 0,1\right] \) consists of elements \(g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \,\right) \) with finite sums of functions \(x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{n}e_{n}\) bounded uniformly in \(0\le t\le 1\) and \(n\ge 1\), but not necessarily convergent function series \(x_{1}e_{1}+x_{2}e_{2}+\cdots \)
Furthermore, denote
$$\begin{aligned} S_{c}\left[ 0,1\right] =&\left\{ g=\left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \in S_{b}\left[ 0,1\right] {:}\,\text {the series} \sum \limits _{n}x_{n}e_{n} \right. \\&\left. \qquad \text {converges in norm }\left\| \cdot \right\| \right\} . \end{aligned}$$
It can be shown that \(S_{c} [0,1]\) as a set of functions is identically equal to C[0, 1], and on account of the well-known open mapping theorem, the norm \(\left\| \cdot \right\| \) is equivalent to \(\Vert \cdot \Vert _{\infty }\); therefore, \(\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right) \) treated as a function Banach space with its norm topology is the same as \(\left( C[0,1], \Vert \cdot \Vert _{\infty }\right) \). Since finite linear combinations \(\sum \nolimits _{k=1}^{n}w_{k}e_{k}\) with rational coefficients form a countable dense set in C[0, 1], the space \(S_{c}\left[ 0,1\right] \) is separable. The Schauder basis \(\left\{ e_{n} ( t ), n\ge 1\right\} \) in \(\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right) \) is monotone, because finite linear combinations of hat functions \(e_{n} ( t ),\,n\ge 1,\) are piecewise linear with an increasing number of nodes. However, the basis \(\left\{ e_{n} ( t ), n\ge 1\right\} \) is not boundedly complete, for bounded finite linear combinations of basic functions need not define a conditionally uniformly convergent series of functions. Moreover, the basis \(\left\{ e_{n} ( t ), n\ge 1\right\} \) in \(S_{c}\left[ 0,1\right] \) is quasi-orthogonal with respect to \(\left\| \cdot \right\| \), but it is not quasi-orthogonal in \( \left( C[0,1], \Vert \cdot \Vert _{\infty }\right) \), and it is not unconditional. More precisely, properties of a given Schauder basis in C[0, 1] depend on the shape of the unit sphere, and from our considerations it follows that for each Schauder basis \(\left\{ e_{n} ( t ), n\ge 1\right\} \) there can be defined a norm \(\left\| \cdot \right\| \) equivalent to the original supremum norm \(\Vert \cdot \Vert _{\infty }\) in C[0, 1], such that the basis \(\left\{ e_{n}\left( t\right) ,n\ge 1\right\} \) becomes quasi-orthogonal with respect to \(\left\| \cdot \right\| \), although the same basis need not be quasi-orthogonal with respect to \(\Vert \cdot \Vert _{\infty }\).
By analogy to \(S_{c,\epsilon }\) one can now define the spaces \(S_{c,\epsilon }\left[ 0,1\right] \) consisting of function series \(\sum _n\epsilon _n x_n e_n\) convergent (conditionally) uniformly in \(0\le t \le 1\) with norms
$$\begin{aligned} \left\| \left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \right\| _{\epsilon } =\left\| \left( \epsilon _{1}x_{1}e_{1}+\epsilon _{2}x_{2}e_{2}+\cdots \right) \right\| , \end{aligned}$$
for all sequences of signs \(\epsilon =\left( \epsilon _{1},\epsilon _{2},\ldots \right) \in \left\{ -1,1\right\} ^{\mathbb {N}}.\) The intersection
$$\begin{aligned} \bigcap \limits _{\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N} }}S_{c,\epsilon }\left[ 0,1\right] :=\ell ^{1}\left[ 0,1\right] \end{aligned}$$
is then the Banach space of function series convergent absolutely uniformly in \(0\le t\le 1,\) with the norm
$$\begin{aligned} \sup \left\{ \left\| \left( x_{1}e_{1}+x_{2}e_{2}+\cdots \right) \right\| _{\epsilon }{:}\,\epsilon \in \left\{ -1,1\right\} ^{\mathbb {N}}\right\} =\sup \limits _{0\le t\le 1}\sum \limits _{n=1}^{\infty }\left| x_{n}e_{n}\left( t\right) \right| . \end{aligned}$$
The last formula follows from the fact that Schauder hat functions \(e_n(t),\,n\ge 1\), are nonnegative, and in a more general case this is a consequence of a theorem by Sierpiński, cf. [10, Prop. 1.5.7, p. 19]. Clearly, the basis \(\{ e_n(t), n\ge 1 \}\) in \(\ell ^{1}\left[ 0,1\right] \) is quasi-orthogonal, monotone, unconditional and boundedly complete; thus, \(\ell ^{1}\left[ 0,1\right] \) possesses the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64 and Ch. III, §3, Corollary 8, p. 83].
To estimate the rate of (uniform) convergence of function series of the form \(g\left( t\right) =\sum \nolimits _{n=1}^{\infty }x_{n}e_{n}\left( t\right) ,\,t\in \left[ 0,1\right] ,\) we may introduce various norms similar as in \(\ell ^{p},\, 1\le p<\infty ;\) namely, let \(\left( R_{n}g\right) \left( t\right) =\sum \nolimits _{k\ge n}x_{k}e_{k} ( t ),\) and let
$$\begin{aligned} S_{c}^{p}\left[ 0,1\right] =\left\{ g =\sum \limits _{n=1}^{\infty }x_{n}e_{n}\left( t\right) \in S_{c}[0,1]{:}\, \left\| g\right\| _{p} =\left( \sum \limits _{~n=1}^{\infty }\left\| \left( R_{n}g\right) \right\| ^{p}\right) ^{1/p} <\infty \right\} . \end{aligned}$$
Arguing as in example 2, we conclude that the basis \(\left\{ e_{n} ( t), n\ge 1\right\} \) in the space \(\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right) \) is quasi-orthogonal, but it is not unconditional. Moreover, the basis \(\left\{ e_{n} (t), n\ge 1\right\} \) in \(\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right) \) is monotone and boundedly complete; thus, by Dunford’s theorem the spaces \(\left( S_{c}^{p}\left[ 0,1\right] ,\left\| \cdot \right\| _{p}\right) \) possess the RNP, see [4, Ch. III, §1, Th. 6, p. 64].

Remark 1

The same idea as above leads to other examples of Banach spaces of a similar kind. For example, let \(I=\left\{ i_{n},n\ge 1\right\} \) be a sequence of positive integers such that \(1\le i_{n}\le n,\,i_{n}\nearrow \infty \) and \(n-i_{n}\nearrow \infty \) as \(n\rightarrow \infty ,\) and for \( x=\left\{ x_{n}\right\} \in \mathbb {R}^{\infty },\) let
$$\begin{aligned} \left\| x\right\| _{I,p}=\left\{ \sum \limits _{n=1}^{\infty }\left( \left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right| \right) ^{p}\right\} ^{1/p}, \end{aligned}$$
where \(1\le p<\infty ,\) and
$$\begin{aligned} \left\| x\right\| _{I,\infty }=\sup \limits _{n\ge 1}\left\{ \left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right| \right\} . \end{aligned}$$
Define
$$\begin{aligned} \mathbb {B}_{I,p}=\left\{ x=\left\{ x_{n}\right\} \in \mathbb {R}^{\infty }{:}\, \text { the series }\sum \limits _{n}x_{n}e_{n}\text { converges in norm } \left\| \cdot \right\| _{I,p}\,\right\} , \end{aligned}$$
\(1\le p\le \infty .\) Then \(\mathbb {B}_{I,p},\,1\le p\le \infty ,\) are Banach spaces such that \(\left\{ e_{n},n\ge 1\right\} \) is a quasi-orthogonal basis with respect to \(\left\| \cdot \right\| _{I,p},\) but in general not unconditional. Moreover, for \(1\le p<\infty \) the basis \( \left\{ e_{n},n\ge 1\right\} \) in \(\mathbb {B}_{I,p}\) is boundedly complete, and thus, these spaces possess the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64]. Next, if \(I^{\left( 1\right) }=\left\{ i_{n}^{\left( 1\right) },n\ge 1\right\} ,\,I^{\left( 2\right) }=\left\{ i_{n}^{\left( 2\right) },n\ge 1\right\} , \ldots , I^{\left( k\right) }=\left\{ i_{n}^{\left( k\right) },n\ge 1\right\} \) are some sequences of positive integers satisfying conditions: \(1\le i_{n}^{\left( 1\right) }<\cdots<i_{n}^{\left( k\right) }\le n,\,i_{n}^{\left( j\right) }\nearrow \infty ,\,1\le j\le k,\, i_{n}^{\left( j+1\right) }-i_{n}^{\left( j\right) }\nearrow \infty ,\,1\le j<k,\) and \(n-i_{n}^{\left( k\right) }\nearrow \infty \) as \(n\rightarrow \infty ,\) then for \(n-i_{n}\ge k\ge 2\) (replacing \(i_{n}\) by \( i_{n}^{\left( 1\right) }\)), the sums \(\left| x_{i_{n}}+\cdots +x_{n}\right| +\left| x_{n+1}\right| \) can be divided into blocks of the form
$$\begin{aligned} \left| x_{i_{n}^{\left( 1\right) }}+\cdots +x_{i_{n}^{\left( 2\right) }-1}\right| +\left| x_{i_{n}^{\left( 2\right) }}+\cdots +x_{i_{n}^{\left( 3\right) }-1}\right| +\cdots +\left| x_{i_{n}^{\left( k\right) }}+\cdots +x_{n}\right| +\left| x_{n+1}\right| ; \end{aligned}$$
in addition, for \(i\ge 1\), instead of \(x_i e_i\), the terms \(\epsilon _i x_i e_i\) can be used, where \(\{ \epsilon _i, i\ge 1 \}\) are various sequences of signs \(\pm \,1\). By means of these expressions, in an analogous way as above, new norms and new Banach spaces with quasi-orthogonal bases can be defined.

4 Convergence of the Banach Random Walk

According to the standard terminology, the measure G on the Borel \(\sigma \)-field \(\mathcal { B}\left( \mathbb {R}\right) \) is called here symmetric, if \(G\left( -A\right) =G\left( A\right) \) for all \(A\in \mathcal {B}\left( \mathbb {R}\right) \).

Lemma 1

Let \(\left\{ b_{n},n\ge 1\right\} \) be a quasi-orthogonal Schauder basis in a real Banach space \(\left( \mathbb {B} ,\left\| \cdot \right\| \right) \), and let \(\left\{ G_{n},n\ge 1\right\} \) be a sequence of symmetric probability distributions concentrated on the interval \(\left[ -\,1,1\right] \subset \mathbb {R}\). Then the BRW \(\left\{ Z_{n},n\ge 1\right\} \) is a \(\mathbb {B}\)-valued martingale with respect to the natural filtration \(\mathcal {F}_{n}=\sigma \left( X_{1},X_{2},\ldots ,X_{n}\right) =\sigma \left( Z_{1},Z_{2},\ldots ,Z_{n}\right) ,\, n\ge 1\).

Proof

From the construction of BRW in a Banach space, it follows that \(Z_{n}\in B\) for all \(n\ge 1\), i.e., the r.e.’s \(Z_{n}\) are bounded and therefore Bochner integrable. Moreover, each \(Z_{n}\) is \(\mathcal {F}_{n}\)-measurable; thus,
$$\begin{aligned} E\left[ Z_{n+1}|\mathcal {F}_{n}\right] =E\left[ Z_{n}+X_{n+1}b_{n+1}| \mathcal {F}_{n}\right] =Z_{n}+E\left[ X_{n+1}|\mathcal {F}_{n}\right] b_{n+1} ~\, \text {a.s.} \end{aligned}$$
Hence it suffices to show the equality \(E\left[ X_{n+1}|\mathcal {F}_{n}\right] =0\) a.s. But the last statement is obvious, since it is known that \(X_{n+1}\) possesses a symmetric distribution in the a.s. symmetric interval \(\left[ \alpha _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right] =\left[ -\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right] \). \(\square \)

Lemma 2

Let \(\left\{ Z_{n},n\ge 1\right\} \) be the BRW in a Banach space \(\mathbb {B}\) with a quasi-orthogonal Schauder basis \(\left\{ b_{n},n\ge 1\right\} \) and let \(\left\{ \mathcal {F}_{n},n\ge 1\right\} \) be the defined above filtration associated with \(\left\{ Z_{n},n\ge 1\right\} .\) Denote by \(\mathcal {F}_{\infty }\) the \(\sigma \)-field generated by the field \(\bigcup _{n=1}^{\infty }\mathcal {F}_{n}\), i.e., \( \mathcal {F}_{\infty }=\sigma \left( \bigcup _{n=1}^{\infty } \mathcal {F}_{n}\right) \). Then the following statements are true:
  1. (a)
    There exists a vector measure \(\upsilon {:}\,\mathcal {F}_{\infty }\rightarrow \mathbb {B}\) with bounded variation, absolutely continuous with respect to P,  such that
    $$\begin{aligned} \left\langle Z_{n},x^{*}\right\rangle \rightarrow \frac{\mathrm{d}\left\langle \upsilon ,x^{*}\right\rangle }{\mathrm{d}P} ~ \text { a.s. for all } x^{*}\in \mathbb {B}^{*}, \end{aligned}$$
     
  2. (b)

    If there is a r.e. \(\xi \in L^{1}\left( \mathcal {F}_{\infty };\mathbb {B} \right) \) such that

     
$$\begin{aligned} \left\langle Z_{n},x^{*}\right\rangle \rightarrow \left\langle \xi ,x^{*}\right\rangle ~ \text { a.s.} \end{aligned}$$
for each \(x^{*}\in \mathbb {B}^{*}\), then
$$\begin{aligned} \Vert Z_{n}-\xi \Vert \rightarrow 0 ~ \text { a.s.} \end{aligned}$$

Proof

Since the BRW \(\left\{ Z_{n},n\ge 1\right\} \) satisfies the condition \( Z_{n}\in B,\,n\ge 1\), we have \(\sup _{n\ge 1}E\left\| Z_{n}\right\| \le 1<\infty .\) Therefore, our result is a direct consequence of a theorem given by Stegall, which can be found in [11, Ch. II, §4.3, Prop. 4.3, p. 132]. \(\square \)

Lemma 3

For each set \(A\in \mathcal {F}_{\infty }\) there exists
$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A}Z_{n}\mathrm{d}P=V\left( A\right) \end{aligned}$$
in the strong topology of \(\mathbb {B}\), and the mapping \(V{:}\,\mathcal {F} _{\infty }\rightarrow \mathbb {B}\) is a countably additive vector measure.

Proof

Observe first that in view of Jensen’s inequality for conditional expectations in a Banach space, the sequence \(\left\{ \left\| Z_{n}\right\| ,\mathcal {F} _{n},n\ge 1\right\} \) is a real-valued submartingale, cf. [11, Ch. II, §4.1, (g), p. 127], or [12]. Furthermore,
$$\begin{aligned} \sup _{n\ge 1}E\left\| Z_{n}\right\| ^{p}\le 1<\infty ~ \text { for each } 1\le p<\infty , \end{aligned}$$
so that r.v.’s \(\left\{ \left\| Z_{n}\right\| ,n\ge 1\right\} \) are uniformly integrable, which implies a.s. convergence \(\left\| Z_{n}\right\| \rightarrow Z_{\infty }\) (and in \(L^{1}\)), where \(Z_{\infty }\in L^{p}=L^{p}(\mathbb {R})\) for every fixed \(1\le p<\infty \), see, e.g., [8, Ch. IV, Th. IV-1-2, p. 62, and Prop. IV-5-24, p. 91]. In particular,
$$\begin{aligned} \int \nolimits _{A}\left\| Z_{n}\right\| \mathrm{d}P\rightarrow \int \nolimits _{A} Z_{\infty } \mathrm{d}P \end{aligned}$$
for each measurable set \(A\in \mathcal {F}\). Next, if \(B\in \bigcup \nolimits _{n}\mathcal {F} _{n}\), then by the martingale property of \(\left\{ Z_{n},n\ge 1\right\} ,\)
$$\begin{aligned} \int \nolimits _{B}Z_{n}\mathrm{d}P\rightarrow V\left( B\right) \end{aligned}$$
strongly in \(\mathbb {B}\). Let \(\varepsilon >0\) be arbitrary and let \(\delta >0\) be chosen in such a way that \(\int \nolimits _{C}Z_{\infty } \mathrm{d}P<\varepsilon /3\) whenever \(C\in \mathcal {F}_{\infty }\) and \(P\left[ C\right] <\delta \). Given any set \(A\in \mathcal {F}_{\infty },\) select \(B\in \bigcup \nolimits _{n} \mathcal {F}_{n}\) satisfying condition \(P\left[ A\div B\right] <\delta \). Obviously,
$$\begin{aligned}&\left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| \le \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| \\&\quad +\,\left\| \int \nolimits _{B}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{m}\mathrm{d}P\right\| +\left\| \int \nolimits _{B}Z_{m}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| . \end{aligned}$$
Moreover,
$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| \le \int \nolimits _{A\setminus B}\left\| Z_{n}\right\| \mathrm{d}P+\int \nolimits _{B\setminus A}\left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{A\div B}\left\| Z_{n}\right\| \mathrm{d}P. \end{aligned}$$
Passing to the limit as \(n\rightarrow \infty \), we obtain
$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A\div B}\left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{A\div B} Z_{\infty } \mathrm{d}P<\varepsilon /3, \end{aligned}$$
thus
$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{n}\mathrm{d}P\right\| <\varepsilon /3 \end{aligned}$$
for sufficiently large \(n\ge n_{0}\). Since the sequence \(\left\{ \int \nolimits _{B}Z_{n}\mathrm{d}P,n\ge 1\right\} \) is Cauchy in \(\mathbb {B}\), we also conclude that
$$\begin{aligned} \left\| \int \nolimits _{B}Z_{n}\mathrm{d}P-\int \nolimits _{B}Z_{m}\mathrm{d}P\right\| <\varepsilon /3 \end{aligned}$$
for all large enough \(m>n\ge n_{1}\). Consequently,
$$\begin{aligned} \left\| \int \nolimits _{A}Z_{n}\mathrm{d}P-\int \nolimits _{A}Z_{m}\mathrm{d}P\right\| <\varepsilon , \end{aligned}$$
whenever \(m>n\ge \max \left\{ n_{0},n_{1}\right\} \). In other words, the sequence of integrals \(\left\{ \int \nolimits _{A}Z_{n}\mathrm{d}P,n\ge 1\right\} \) is Cauchy in \(\left( \mathbb {B},\left\| \cdot \right\| \right) \), and therefore, there exists
$$\begin{aligned} \lim _{n\rightarrow \infty }\int \nolimits _{A}Z_{n}\mathrm{d}P=V\left( A\right) \end{aligned}$$
in the strong topology of \(\mathbb {B}\) for each set \(A\in \mathcal {F}_{\infty }\). It can be easily seen that \(V{:}\, \mathcal {F}_{\infty } \rightarrow \mathbb {B}\) is finitely additive. Let \(A_{1},A_{2},\ldots \in \mathcal {F}_{\infty }\) be an arbitrary sequence of pairwise disjoint sets. Notice that
$$\begin{aligned} \left\| \lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=1}^{\infty }A_{k}}Z_{n}\mathrm{d}P-\lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=1}^{m}A_{k}}Z_{n}\mathrm{d}P\right\| =\lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}}Z_{n}\mathrm{d}P\right\| , \end{aligned}$$
and thus, to prove countable additivity of V it is enough to show that
$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}}Z_{n}\mathrm{d}P\right\| \rightarrow 0 \end{aligned}$$
as \(m\rightarrow \infty \). Taking \(m_{0}\) so large that \(P\left[ \bigcup \nolimits _{k=m+1}^{\infty }A_{k}\right] <\delta \) for \(m\ge m_{0}\), we obtain
$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} Z_{n}\mathrm{d}P\right\| \le \lim _{n\rightarrow \infty }\int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} \left\| Z_{n}\right\| \mathrm{d}P=\int \nolimits _{\bigcup \nolimits _{k=m+1}^{\infty }A_{k}} Z_{\infty } \mathrm{d}P<\varepsilon /3 \end{aligned}$$
provided \( m\ge m_{0} \), which terminates the proof. \(\square \)

The above Lemma 3 enables us to apply the Lebesgue decomposition theorem for the vector measure V, see [4, Ch. I, §5, Th. 9, p. 31].

Lemma 4

Let
$$\begin{aligned} V\left( A\right) =\lim _{n\rightarrow \infty }\int _{A}Z_{n}\mathrm{d}P, ~\, A\in \mathcal {F}_{\infty }, \end{aligned}$$
and let \(V=H+J,\,\left| H\right| \ll P,\,\left| J\right| \perp P,\) be the Lebesgue decomposition of  V with respect to P, where \( \left| H\right| ,\,\left| J\right| \) are variations of H and J, respectively. Then \(\lim _{n\rightarrow \infty }Z_{n}\) exists a.s. if and only if H has a Radon–Nikodym derivative \(h\in L^{1}\left( \mathcal {F};\mathbb {B}\right) .\) Moreover, in this case \( \lim _{n\rightarrow \infty }Z_{n}=E\left( h|\mathcal {F}_{\infty }\right) \) a.s.

Proof

Arguing as above, we easily note that \(\left\{ Z_{n},\mathcal {F}_{n},n\ge 1\right\} \) is an \(L^{1}\left( \mathbb {B}\right) \)-bounded martingale (here, and in the sequel \(L^{p}\left( \mathbb {B}\right) =L^{p}\left( \varOmega , \mathcal {F}, P ;\mathbb {B}\right) ,\,1\le p < \infty \)); thus, the conclusion follows from the martingale pointwise convergence theorem given in [4, Ch. V, §2, Th. 9, p. 130]. \(\square \)

Lemma 5

Let \(\mathbb {B}\) be a Banach space with the RNP and a quasi-orthogonal Schauder basis \(\left\{ b_{n},n\ge 1\right\} \). Moreover, let \(\left\{ G_{n},n\ge 1\right\} \) be a sequence of symmetric probability distributions concentrated on \(\left[ -\,1,1\right] \subset \mathbb {R}\). Then the BRW martingale \(\left\{ Z_{n},n\ge 1\right\} \) converges strongly a.s. and in \(L^{p}\left( \mathbb {B}\right) \) for each fixed \(1\le p<\infty \).

Proof

The limit \(\lim _{n\rightarrow \infty }Z_{n}\) of the martingale \(\left\{ Z_{n},n\ge 1\right\} \) exists in \(L^{p}\left( \mathbb {B}\right) \)-norm, if and only if \( \sup _{n\ge 1}\left\| Z_{n}\right\| _{p}^{p}=\sup _{n\ge 1}E\left\| Z_{n}\right\| ^{p}<\infty ,\) where \(1<p<\infty ,\) which is evident as \(Z_{n}\in B\) for \(n\ge 1.\) The last observation implies uniform integrability of random elements \(\left\{ Z_{n},n\ge 1\right\} ,\) and we have obviously \(\sup _{n\ge 1}\left\| Z_{n}\right\| _{1}=\sup _{n\ge 1}E\left\| Z_{n}\right\| \le 1<\infty .\) Thus \(\lim _{n\rightarrow \infty }Z_{n}\) exists as well in \(L^{1}\left( \mathbb {B}\right) \)-norm in view of the martingale mean convergence theorem, cf. [4, Ch. V, §2, Corollary 4, p. 126]. It is also well known that an \(L^{1}\left( \mathbb {B}\right) \) convergent martingale converges a.s. to its \(L^{1}\left( \mathbb {B}\right) \)-limit, see [4, Ch. V, §2, Th. 8, p. 129], or [11, Ch. II, §4.3, Th. 4.2, p. 131 and Th. 4.3, p. 136]. \(\square \)

Corollary 1

If \(\mathbb {B}\) is a Banach space with a quasi-orthogonal boundedly complete Schauder basis \(\left\{ b_{n},n\ge 1\right\} ,\) then the BRW \(\left\{ Z_{n},n\ge 1\right\} \) in \(\mathbb {B}\) converges strongly a.s. and in \(L^{p}\left( \mathbb {B}\right) \) for each fixed \(1\le p<\infty \).

Proof

By a theorem of Dunford, if a Banach space \(\mathbb {B}\) possesses a boundedly complete Schauder basis, then \(\mathbb {B}\) has the RNP, cf. [4, Ch. III, §1, Th. 6, p. 64]. Hence and from Lemma 5, the assertion of Corollary 1 follows. \(\square \)

Corollary 2

Let \(\mathbb {B}\) be a reflexive Banach space which has a quasi-orthogonalSchauder basis \(\left\{ b_{n},n\ge 1\right\} \). Then the \(\mathbb {B}\)-valued BRW \(\left\{ Z_{n},n\ge 1\right\} \) converges strongly a.s. and in \(L^{p}\left( \mathbb {B}\right) \) for each fixed \(1\le p<\infty \). In particular, if \(\mathbb {B}=\mathbb {H}\) is a Hilbert space with a basis \(\left\{ b_{n},n\ge 1\right\} \) which forms a CONS in \(\mathbb {H}\), then the last statement remains valid.

Proof

It is fairly well known from a theorem of Phillips that reflexive Banach spaces have the RNP, see [4, Ch. III, §3, Corollary 4, p. 82]. Since each Hilbert space is reflexive, we conclude that \(\mathbb {B}=\mathbb {H}\) has the RNP. Thus, an application of Lemma 5 concludes the proof. \(\square \)

Theorem 1

Let \(\phi {:}\,\mathbb {B}\rightarrow \mathbb {R}\) be a bounded and continuous mapping in a Banach space \(\mathbb {B}\) which has the RNP and a quasi-orthogonal Schauder basis \(\left\{ b_{n},n\ge 1\right\} \). If \( \left\{ Z_{n},n\ge 1\right\} \) is the BRW in \(\mathbb {B},\) then
$$\begin{aligned} \phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right) ~\, \text {a.s. and in } L^{p}=L^{p}\left( \mathbb {R}\right) ,\quad 1\le p<\infty , \end{aligned}$$
where \(\xi =\lim _{n\rightarrow \infty }Z_{n}\) a.s. and in \(L^{p}\left( \mathbb {B}\right) \)-norm for all \(1\le p<\infty .\) In particular, there exists
$$\begin{aligned} \lim _{n\rightarrow \infty }E\phi \left( Z_{n}\right) =E\phi \left( \xi \right) . \end{aligned}$$

Proof

Since \(\phi \) is continuous and the assumptions of Lemma 5 are fulfilled, we conclude that \(\phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right) \) a.s. But in addition \(\phi \) is assumed to be bounded, thus using the Lebesgue-dominated convergence theorem we obtain also convergence \( \phi \left( Z_{n}\right) \rightarrow \phi \left( \xi \right) \) in \( L^{p},\,1\le p<\infty \). The last statement of the theorem follows from the estimate
$$\begin{aligned} \left| E\phi \left( Z_{n}\right) -E\phi \left( \xi \right) \right| \le E\left| \phi \left( Z_{n}\right) -\phi \left( \xi \right) \right| \rightarrow 0. \end{aligned}$$
\(\square \)

Corollary 3

The assertion of Theorem 1 remains valid for a Banach space \( \mathbb {B}\) with a boundedly complete quasi-orthogonal Schauder basis \( \left\{ b_{n},n\ge 1\right\} ,\) as well as for a Hilbert space \(\mathbb {B}=\mathbb {H}\) with the Schauder basis \(\left\{ b_{n},n\ge 1\right\} \) that forms a CONS in \(\mathbb {H}\).

5 The Banach Functional Integral in a Class of Banach Spaces

Let \(\mathcal {C}_b =\left\{ \phi {:}\,B\rightarrow \mathbb {R};\phi \text {-bounded and continuous}\right\} \). Observe that \(\mathcal {C}_b\) has the following properties:

(i\(_{1}\)) the set \(\mathcal {C}_b\) is a real linear space,

(i\(_{2}\)) if \(\phi \in \mathcal {C}_b\), then \(\left| \phi \right| \in \mathcal {C}_b\).

Define a functional \(f{:}\,\mathcal {C}_b\rightarrow \mathbb {R}\) by the formula:
$$\begin{aligned} f\left( \phi \right) =\lim _{n\rightarrow \infty }E\phi \left( Z_{n}\right) =E\phi \left( \xi \right) , \end{aligned}$$
(4)
where \(\left\{ Z_{n},n\ge 1\right\} \) is a BRW in the Banach space \(\mathbb { B} \), and \(\xi =\lim _{n\rightarrow \infty }Z_{n}\) a.s. and in \( L^{p}\left( \mathbb {B}\right) ,\,1\le p<\infty .\)

It can be easily seen that the mapping f satisfies the following conditions:

(ii\(_{1}\)) \(f{:}\,\mathcal {C}_b\rightarrow \mathbb {R}\) is a linear functional,

(ii\(_{2}\)) the functional f is nonnegative, i.e., \(f\left( \phi \right) \ge 0\) whenever \(\phi \in \mathcal {C}_b\) and \(\phi \ge 0\),

(ii\(_{3}\)) if 1\(^{0}\,\left\{ \phi _{n}\right\} \subset \mathcal {C }_b,\,\psi \in \mathcal {C}_b\), 2\(^{0}\,\left| \phi _{n}\right| \le \psi \) for \(n\ge 1\), and 3\(^{0}\,\lim _{n\rightarrow \infty }\phi _{n}\left( x\right) =0\) for all \(x\in B\), then \(\lim _{n\rightarrow \infty }f\left( \phi _{n}\right) =0\).

Notice that 3\(^{0}\) implies P-a.s. convergence \(\phi _{n}\left( \xi \right) \rightarrow 0\); thus, the last condition follows from the classical Lebesgue-dominated convergence theorem applied to integrals \(E\phi _{n}\left( \xi \right) ,\,n=1,2,\ldots \) (Actually, in our approach we can even replace condition 3\(^{0}\) by a weaker assumption \(\phi _{n} \rightarrow 0\) in \(P\circ \xi ^{-1}\)-measure.) Consequently, the functional f satisfies all the conditions given in §2 of the Banach paper [1]. Therefore, for our functional f the Banach Th. 1, §3, p. 322, [1] is valid. In this way, we obtain the following result.

Theorem 2

Let \(\left\{ Z_{n},n\ge 1\right\} \) be a BRW in a Banach space \( \mathbb {B}\) with the RNP and a quasi-orthogonal Schauder basis \( \left\{ b_{n},n\ge 1\right\} ,\) in particular in a Banach space \( \mathbb {B}\) with a boundedly complete quasi-orthogonal Schauder basis. Then, the functional f given by (4) has an extension to the additive functional F on the linear set \(\mathcal {L}\supset \mathcal {C}_b\) of all bounded, Borel measurable functions \(\varPhi {:}\, B\rightarrow \mathbb {R}\). Moreover, the extended functional F on \(\mathcal {L}\) possesses all the properties (A)–(E) and (R) specified in §1 of the Banach paper [1], analogous to the Lebesgue integral.

Remark 2

The approach presented above is a generalization of the method proposed by Banach [1] for the construction of the so-called \(\mathfrak {L}\)-integral—an analogue of the Lebesgue integral in abstract spaces.

Theorem 3

Let \(\mathbb {B}\) be a Banach space with the RNP and a quasi-orthogonal Schauder basis, in particular—a Banach space with a boundedly complete quasi-orthogonal Schauder basis. Then each sequence of symmetric probability distributions \(\left\{ G_{n},n\ge 1\right\} \) concentrated on the interval \( [-\,1,1]\subset \mathbb {R}\) generates a probability measure \(\varGamma \) on the Borel \(\sigma \)-field \(\mathcal {B}\) in \(\mathbb {B},\) given by
$$\begin{aligned} \varGamma \left( A\right) =E\mathbb {1}_{A}\left( \xi \right) , ~ A\in \mathcal {B}. \end{aligned}$$
The measure \(\varGamma \) is equal to the limit distribution of the described above BRW \(\left\{ Z_{n},n\ge 1\right\} \) in \(\mathbb {B}\), thus \({\mathrm{supp}\,}\varGamma \subseteq B.\)

Proof

Obviously, \(\varGamma \) is nonnegative and normalized so that \(\varGamma \left( \mathbb {B}\right) =\varGamma \left( B\right) =1\). It suffices to verify countable additivity of \(\varGamma \), but it follows immediately from the properties of the integral \(E\left( \cdot \right) \). The last conclusion can also be easily shown in a direct way. To this end, let \(A_{1},A_{2},\ldots \in \mathcal {B}\) be arbitrary disjoint sets. Since \(\mathrm {supp\,}\varGamma \subseteq B,\) we have \(\varGamma \left( A \right) =\varGamma \left( A \cap B\right) =E\mathbb {1}_{A \cap B}\left( \xi \right) ,\,A \in \mathcal {B}\). Observe next that
$$\begin{aligned} E\mathbb {1} _{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) = E\left( \sum \limits _{j=1}^{n}\mathbb {1}_{A_{j}\cap B}\left( \xi \right) \right) =\sum \limits _{j=1}^{n}E\mathbb {1}_{A_{j}\cap B}\left( \xi \right) =\sum \limits _{j=1}^{n}\varGamma \left( A_{j}\right) , \end{aligned}$$
and
$$\begin{aligned} 0\le \sum \limits _{j=1}^{n}\mathbb {1}_{A_{j}\cap B}\left( x\right) = \mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( x\right) \nearrow \mathbb {1}_{\bigcup \nolimits _{n=1}^{\infty }\left( A_{n}\cap B\right) }\left( x\right) \le \mathbb {1}_{B}\left( x\right) ,~ x\in B. \end{aligned}$$
Hence, on account of the Lebesgue monotone convergence theorem,
$$\begin{aligned} \sum \limits _{n=1}^{\infty }\varGamma \left( A_{n}\right)= & {} \lim _{n\rightarrow \infty }\sum \limits _{j=1}^{n}\varGamma \left( A_{j}\right) =\lim _{n\rightarrow \infty }E\mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) \\= & {} E\left( \lim _{n\rightarrow \infty }\mathbb {1}_{\bigcup \nolimits _{j=1}^{n}\left( A_{j}\cap B\right) }\left( \xi \right) \right) =E\mathbb {1}_{\left( \bigcup \nolimits _{n=1}^{\infty } A_{n}\right) \cap B }\left( \xi \right) =\varGamma \left( \bigcup \limits _{n=1}^{\infty }A_{n}\right) . \end{aligned}$$
\(\square \)
From the construction of the BRW in a Banach space, it follows immediately that the limit distribution \(\varGamma =P\circ \xi ^{-1}\) of the BRW is sign-invariant with respect to the Schauder basis \(\{ b_{n}, n\ge 1\}\), in the sense that for each set \(A\in \mathcal {B}\) and every sequence \(\epsilon =\{\epsilon _{1},\epsilon _{2},\ldots \}\) of signs \(\epsilon _{k} \in \{-1,1\},\,k\ge 1\), we have
$$\begin{aligned} \varGamma (A) = \varGamma (\epsilon A), \end{aligned}$$
where \(\epsilon A= \left\{ \sum _{k=1}^{\infty } \epsilon _{k} x_{k} b_{k} \in \mathbb {B} {:}\, \sum _{k=1}^{\infty } x_{k} b_{k} \in A\right\} \). It is also clear that each sign-invariant measure is symmetric, thus
$$\begin{aligned} \varGamma (A) = \varGamma (-A) ~\, \text {for all} ~ A\in \mathcal {B}. \end{aligned}$$
By analogy to the notion of the Wiener measure, we propose to call \(\varGamma \)the Banach measure in a Banach space. One may expect that the Banach measure will play a similarly important role in Banach spaces as is played by the Gaussian measure constructed by Gross [5], cf. Bogachev [3], or Kuo [6].

6 Examples

  1. 1.
    Let \(\mathbb {B}=\ell ^{p},\,1\le p < \infty \), and let \(b_n = e_n = (0,\ldots ,0,1,0,\ldots ),\,n\ge 1\), where 1 is the nth term of the sequence \((0,\ldots ,0,1,0,\ldots )\). Then \(\pi _n(x) = \sum _{k=1}^{n} x_k b_k = (x_{1},\ldots ,x_{n},0,\ldots )\) for \(x=\sum _{n=1}^{\infty }x_{n}b_{n} = (x_{1},x_{2},\ldots ) \in \ell ^{p} \), and thus
    $$\begin{aligned} \left| \pi _{n}(x)+tb_{n+1}\right| _{p}^{p}=\sum \limits _{k=1}^{n}|x_{k}|^{p}+|t|^{p}\le 1 ~ \Leftrightarrow ~ |t|\le \left( 1-\sum \limits _{k=1}^{n}|x_{k}|^{p}\right) ^{1/p}. \end{aligned}$$
     
Hence it follows that \(\{b_{n},n\ge 1\}\) is a quasi-orthogonal Schauder basis in \(\ell ^{p}\). (In fact, the considered basis is unconditional.) Moreover, if
$$\begin{aligned} \sup _{n\ge 1}\left| \pi _{n}(x)\right| _{p}=\sup _{n\ge 1}\left| \sum \limits _{k=1}^{n}x_{k}b_{k}\right| _{p}=\left( \sum \limits _{k=1}^{\infty }|x_{k}|^{p}\right) ^{1/p}\le M<\infty , \end{aligned}$$
then the series \(\sum _{n=1}^{\infty }x_{n}b_{n}\) converges in \(\ell ^{p}\). Thus, the basis \(\{b_{n},n\ge 1\}\) is boundedly complete and in consequence each space \(\ell ^{p},\,1\le p < \infty \), has the RNP. It is also well known that for \(p >1 \) the spaces \(\ell ^p\) are reflexive, which implies as well that they have the RNP. Therefore, all the above results are valid for Banach spaces \(\mathbb {B}=\ell ^{p},\,1\le p < \infty \).
  1. 2.
    Let \(\mathbb {B}=L^{p}\left[ 0,1\right] ,\) where \(1\le p<\infty .\) Consider the system of Haar functions: \(h_{1}^{0}\left( s\right) =1 ,\,s\in \left[ 0,1\right] \), and
    $$\begin{aligned} h_{k}^{n}\left( s\right) = 2^{n}\cdot \mathbb {1}_{[\left( 2k-2\right) /2^{n+1},\,\left( 2k-1\right) /2^{n+1})}\left( s\right) - 2^{n}\cdot \mathbb {1}_{[\left( 2k-1\right) /2^{n+1},\,\left( 2k\right) /2^{n+1})}\left( s\right) \end{aligned}$$
     
for \(k=1,2,\ldots ,2^{n},\,n=1,2,\ldots ,\,s\in \left[ 0,1\right] \). It is known that the system of Haar functions forms a Schauder basis in \(L^{p}\left[ 0,1\right] \), see, e.g., [9, Th. 24.17, pp. 290–295], or [7, Part II, Prop. 2.c.1, p. 150]. For convenience of the reader, we sketch here the proof that the Haar basis in \(L^{p}\left[ 0,1\right] \) is quasi-orthogonal.

Proposition 1

The system of Haar functions is a quasi-orthogonal basis in \(L^{p}\left[ 0,1\right] ,\,1\le p<\infty \).

Proof

Let the Haar functions be arranged in a sequence that is divided into blocks, each of \(2^{n}\) members, numbered by upper indices \( n=0,1,2,\ldots , \)
$$\begin{aligned} \varLambda =\left\{ \left( h_{1}^{0}\right) ,\left( h_{1}^{1},h_{2}^{1}\right) ,\left( h_{1}^{2},h_{2}^{2},h_{3}^{2},h_{4}^{2}\right) ,\left( h_{1}^{3},h_{2}^{3},h_{3}^{3},h_{4}^{3},h_{5}^{3},h_{6}^{3},h_{7}^{3},h_{8}^{3}\right) ,\ldots \right\} . \end{aligned}$$
We make now two crucial observations: 1\(^{0}\) for a fixed \(n\ge 0\) within the same nth block the Haar functions \(h_{k}^{n},\, 1\le k\le 2^{n} \), have nonoverlapping supports, 2\(^{0}\) the function of the form \( c_{1}^{0}h_{1}^{0}+c_{1}^{1}h_{1}^{1}+c_{2}^{1}h_{2}^{1}+\cdots +c_{1}^{n-1}h_{1}^{n-1}+ \cdots +c_{2^{n-1}}^{n-1}h_{2^{n-1}}^{n-1}, \) where \(c_{j}^{i}\in \mathbb {R}\) are arbitrarily fixed coefficients, is constant in each interval of the form \(\left[ \left( k-1\right) /2^{n},k/2^{n}\right) ,\,k=1,2,\ldots ,2^{n}.\)
Suppose now that \(\pi _{k-1}^{n}\left( x\right) =x_{1}^{0}h_{1}^{0}+x_{1}^{1}h_{1}^{1}+x_{2}^{1}h_{2}^{1}+\cdots +x_{k-1}^{n}h_{k-1}^{n} \) and the next element of \(\varLambda \) is \(h_{k}^{n}.\) Then for any parameter \(t\in \mathbb {R},\)
$$\begin{aligned} \left\| \pi _{k-1}^{n}\left( x\right) +th_{k}^{n}\right\| _{p}^{p}= & {} \int \nolimits _{0}^{1}\left| \pi _{k-1}^{n}\left( x\right) \left( s\right) +th_{k}^{n}\left( s\right) \right| ^{p}\mathrm{d}s=\int \nolimits _{0}^{\left( k-1\right) /2^{n}}\left| \pi _{k-1}^{n}\left( x\right) \left( s\right) \right| ^{p}\mathrm{d}s\\&+\,\int \nolimits _{\left( k-1\right) /2^{n}}^{k/2^{n}}\left| \pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) +th_{k}^{n}\left( s\right) \right| ^{p}\mathrm{d}s\\&+\,\int \nolimits _{k/2^{n}}^{1}\left| \pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) \right| ^{p}\mathrm{d}s=I_{1}+I_{2}+I_{3}, \end{aligned}$$
where \(\pi _{2^{n-1}}^{n-1}\left( x\right) \left( s\right) = \pi _{2^{n-1}}^{n-1}\left( x\right) \left( (2k-1)/2^{n+1}\right) =c ,\,s\in \big [ (k-1)/{2^{n}}, k/{2^{n}}\big ).\) The first and third integrals on the right-hand side do not depend on the parameter t,  and the middle term is equal to
$$\begin{aligned} I_{2}= & {} \int \nolimits _{\left( k-1\right) /2^{n}}^{\left( 2k-1\right) /2^{n+1}}\left| c+t2^{n}\right| ^{p}\mathrm{d}s+\int \nolimits _{\left( 2k-1\right) /2^{n+1}}^{k/2^{n}}\left| c-t2^{n}\right| ^{p}\mathrm{d}s\\= & {} \left| c+t2^{n}\right| ^{p}\cdot \frac{1}{2^{n+1}}+\left| c-t2^{n}\right| ^{p}\cdot \frac{1}{2^{n+1}}:=r\left( t\right) . \end{aligned}$$
Since \(r\left( t\right) =r\left( -t\right) \), and
$$\begin{aligned} \inf \left\{ t\in \mathbb {R}{:}\,r\left( t\right) \le 1-I_{1}-I_{3}\right\}= & {} -\sup \left\{ -t\in \mathbb {R}{:}\,r\left( -t\right) \le 1-I_{1}-I_{3}\right\} \\= & {} -\sup \left\{ t^{\prime }\in \mathbb {R}{:}\,r\left( t^{\prime }\right) \le 1-I_{1}-I_{3}\right\} , \end{aligned}$$
we conclude that \(\alpha _{k}^{n}=-\beta _{k}^{n}\). The same argument remains valid when \(\pi _{k-1}^{n}\) is replaced by \(\pi _{2^{n-1}}^{n-1},\) and \(\pi _{k}^{n}\) is replaced by \(\pi _{1}^{n},\) and thus, the system of Haar functions forms a quasi-orthogonal basis in \(L^{p}\left[ 0,1\right] .\)\(\square \)

Remark 3

It is clear that the Haar functions for \(n\ge 1\) can be modified as follows:
$$\begin{aligned} h_k^n\left( s\right) = 2^{n/p}\cdot \mathbb {1}_{[\left( 2k-2\right) /2^{n+1},\,\left( 2k-1\right) /2^{n+1})}\left( s\right) - 2^{n/p}\cdot \mathbb {1}_{[\left( 2k-1\right) /2^{n+1},\,\left( 2k\right) /2^{n+1})}\left( s\right) , \end{aligned}$$
\(k=1,2,\ldots ,2^{n},\,s\in \left[ 0,1\right] \). Then the above Proposition 1 for modified Haar functions remains true, and in addition, we have \(\Vert h_{k}^{n} \Vert _{p}= 1\) for all kn.

Evidently, all the spaces \(L^{p}\left[ 0,1\right] ,\,1<p<\infty \), are reflexive Banach spaces, and thus, they possess the RNP, which is a straightforward consequence of Phillips’ theorem, cf. [4, Ch. III, §3, Corollary 6, p. 82]. Unfortunately, the space \(L^{1}\left[ 0,1\right] \) does not have the RNP, see [4, Ch. VII, p. 219].

Hence, it follows that all the results presented in previous sections are valid for Banach spaces \(L^{p}[0,1],\,1<p<\infty \).

Remark 4

Proposition 1 together with observation that the space \(L^{1}[0,1]\) does not possess the RNP implies the following conclusion: the existence of a quasi-orthogonal Schauder basis in a Banach space is not a sufficient condition for the RNP. The same conclusion follows from the fact that the space \(\left( S_{c}\left[ 0,1\right] ,\left\| \cdot \right\| \right) \) (isometrically isomorphic to \(\left( C[0,1], \Vert \cdot \Vert _{\infty }\right) \)) does not have the RNP.

3. To illustrate the technique of computations of Banach \(\mathfrak {L}\)-integrals based on the method described here, we calculate, for instance, two “rarefied” absolute pth moments of \(\xi =\sum _{n=1}^{\infty }X_{n}b_{n}\) in \(\ell ^{p},\,1\le p<\infty \), determined by lacunary series \(\sum _{n=1}^{\infty } X_{2n-1} b_{2n-1} \) and \( \sum _{n=1}^{\infty } X_{2n} b_{2n} \), namely \(E\left( \sum _{n=1}^{\infty }\left| X_{2n-1}\right| ^{p}\right) \) and \( E\left( \sum _{n=1}^{\infty }\left| X_{2n}\right| ^{p}\right) \), where \(\xi \) is the limit random element of the BRW generated by the sequence of identical uniform distributions on \(\left[ -\,1,1\right] \subset \mathbb {R}\). Since the basis \(\{ b_n, n\ge 1 \}\) in \(\ell ^{p}\) is unconditional, the lacunary series considered here are strongly convergent, see, e.g., [7, Prop. 1.c.1, p. 15, and p. 19].

Observe first that if \(\pi _\mathrm{odd}\left( x\right) =\sum _{n=1}^{\infty }x_{2n-1}b_{2n-1}\) for \(x=\sum _{n=1}^{\infty }x_{n}b_{n}\in \ell ^{p}, \) then \(\left| \pi _\mathrm{odd}\left( x\right) \right| _{p}=\left( \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p}, \) thus
$$\begin{aligned}&\left| \left( \sum \limits _{~n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p}-\left( \sum \limits _{n=1}^{\infty }\left| x_{2n-1}^{\prime }\right| ^{p}\right) ^{1/p}\right| =\left| \left| \pi _\mathrm{odd}\left( x\right) \right| _{p}-\left| \pi _\mathrm{odd}\left( x^{\prime }\right) \right| _{p}\right| \\&\quad \le \left| \pi _\mathrm{odd}\left( x-x^{\prime }\right) \right| _{p}=\left( \sum \limits _{~n=1}^{\infty }\left| x_{2n-1}-x_{2n-1}^{\prime }\right| ^{p}\right) ^{1/p}\le \left| x-x^{\prime }\right| _{p}\rightarrow 0 \end{aligned}$$
as \(x\rightarrow x^{\prime }\) in \(\ell ^{p}\)-norm \(\left| \cdot \right| _{p}\). It follows that \(\ell ^{p}\ni x\mapsto \left( \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p}\right) ^{1/p}\) is a continuous function. Therefore, \(\ell ^{p}\ni x\mapsto \sum _{n=1}^{\infty }\left| x_{2n-1}\right| ^{p},\) as well as \(\ell ^{p}\ni x\mapsto \sum _{n=1}^{\infty }\left| x_{2n}\right| ^{p} \) are continuous, and both these maps are bounded in the unit ball \(B \subset \ell ^{p} \).
According to the construction of the BRW in \(\ell ^{p},\) the density of \(X_1\) is equal to \(f_1(x_1)=\mathbb {1}_{[-1,1]} (x_1)/2\), and the density of \(\left( X_{1},X_{2},\ldots ,X_{n}\right) \) for \(n>1\) is given by
$$\begin{aligned}&f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \\&\quad =\frac{\mathbb {1}_{K_{n}}\left( x_{1},x_{2},\ldots ,x_{n}\right) }{2^{n}\cdot \left( \left[ 1-\left| x_{1}\right| ^{p}\right] \cdot \left[ 1-\left( \left| x_{1}\right| ^{p}+\left| x_{2}\right| ^{p}\right) \right] \cdot \ldots \cdot \left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] \right) ^{1/p}}, \end{aligned}$$
where \(K_{n}=K_{n}\left( 0,1\right) \) is the unit ball with center zero and radius 1 in \(\mathbb {R}^{n}\), equipped with the \(\ell ^p\)-norm \(\left| \left( x_{1},\ldots ,x_{n}\right) \right| _{n,p}= \left( \sum \nolimits _{k=1}^{n}\left| x_{k}\right| ^{p}\right) ^{1/p}\). Hence,
$$\begin{aligned} E\left| X_{1}\right| ^{p}=\int \nolimits _{-1}^{1}\frac{\left| x_{1}\right| ^{p}}{2}\mathrm{d}x_{1}=2\int \nolimits _{0}^{1}\frac{x_{1}^{p}}{2} \mathrm{d}x_{1}=\left. \frac{x_{1}^{p+1}}{p+1}\right| _{0}^{1}=\frac{1}{p+1}. \end{aligned}$$
Moreover, for \(n > 1\),
$$\begin{aligned} E\left| X_{n}\right| ^{p}= & {} \int \nolimits _{K_{n}}\left| x_{n}\right| ^{p}\cdot f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n}\\= & {} \int \nolimits _{K_{n-1}} \left( 2\int \nolimits _{0}^{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} x_{n}^{p}\cdot f_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) \mathrm{d}x_{n}\right) \\&\times \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \int \nolimits _{K_{n-1}} \left. \frac{x_{n}^{p+1}}{p+1} \right| _{0}^{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} \\&\times \frac{f_{n-1}\left( x_{1},x_{2},\ldots ,x_{n-1}\right) }{\left[ 1-\left( \left| x_{1}\right| ^{p} +\cdots + \left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}} \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \int \nolimits _{K_{n-1}}\frac{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] }{p+1} \cdot f_{n-1}\left( x_{1},x_{2},\ldots ,x_{n-1}\right) \\&\times \mathrm{d}x_{1}\mathrm{d}x_{2}\ldots \mathrm{d}x_{n-1}\\= & {} \frac{1}{p+1}\left\{ 1-\left( E\left| X_{1}\right| ^{p}+E\left| X_{2}\right| ^{p}+\cdots +\left| X_{n-1}\right| ^{p}\right) \right\} . \end{aligned}$$
Thus,
$$\begin{aligned} E\left| X_{2}\right| ^{p}=\frac{1}{p+1}\left\{ 1-E\left| X_{1}\right| ^{p}\right\} =\frac{1}{p+1}\left( 1-\frac{1}{p+1}\right) = \frac{1}{p+1}\frac{p}{p+1}, \end{aligned}$$
and, by induction,
$$\begin{aligned} E\left| X_{n}\right| ^{p}= & {} \frac{1}{p+1}\left\{ 1-\left( \frac{1}{p+1} +\frac{p}{\left( p+1\right) ^{2}}+\cdots +\frac{p^{n-2}}{\left( p+1\right) ^{n-1} }\right) \right\} \\= & {} \frac{1}{\left( p+1\right) }\left\{ 1-\frac{1}{p+1}\cdot \frac{1-p^{n-1}/\left( p+1\right) ^{n-1}}{1-p/\left( p+1\right) }\right\} =\frac{p^{n-1}}{\left( p+1\right) ^{n}}. \end{aligned}$$
Therefore,
$$\begin{aligned} E\left( \sum \limits _{~n=1}^{\infty }\left| X_{2n-1}\right| ^{p}\right) =\sum \limits _{n=1}^{\infty }\frac{p^{2n-1-1}}{\left( p+1\right) ^{2n-1}}=\frac{1}{p+1}\cdot \frac{1}{1-p^{2}/\left( p+1\right) ^{2}}= \frac{p+1}{2p+1}, \end{aligned}$$
and
$$\begin{aligned} E\left( \sum \limits _{~n=1}^{\infty }\left| X_{2n}\right| ^{p}\right) =\sum \limits _{n=1}^{\infty }\frac{p^{2n-1}}{\left( p+1\right) ^{2n}}=\frac{p }{\left( p+1\right) ^{2}}\cdot \frac{1}{1-p^{2}/\left( p+1\right) ^{2}}= \frac{p}{2p+1}. \end{aligned}$$
In consequence,
$$\begin{aligned} E\left| \pi _{\mathrm{odd}}\left( \xi \right) \right| _{p}^{p}=\frac{ p+1}{2p+1},\quad E\left| \xi -\pi _\mathrm{odd}\left( \xi \right) \right| _{p}^{p}=\frac{p}{2p+1}, \end{aligned}$$
so that
$$\begin{aligned} E\left| \xi \right| _{p}^{p}=E\left( \sum \limits _{~n=1}^{\infty }\left| X_{n}\right| ^{p}\right) =\frac{p+1}{2p+1}+\frac{p}{2p+1}=1. \end{aligned}$$

Notes

Acknowledgements

The authors are grateful to the referee and the associate editor for their helpful remarks which led to a noticeable improvement of the paper.

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Authors and Affiliations

  1. 1.Pope John Paul II State School of Higher Education in Biała PodlaskaBiała PodlaskaPoland
  2. 2.Faculty of Mathematics, Informatics and Landscape ArchitectureThe John Paul II Catholic University of LublinLublinPoland

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