Journal of Theoretical Probability

, Volume 32, Issue 1, pp 47–63

# Limit Distribution of the Banach Random Walk

• Patrycja Jędrzejewska
• August M. Zapała
Open Access
Article

## Abstract

We consider various probability distributions $$\{G_n, n\ge 1\}$$ concentrated on the interval $$[-1,1]\subset \mathbb {R}$$ and investigate basic properties of the limit distribution $$\Gamma$$ of the Banach random walk in a Banach space $$\mathbb {B}$$ generated by $$\{G_n , n\ge 1\}$$. In particular, we describe assumptions ensuring that the support of $$\Gamma$$ is equal to the unit sphere in $$\mathbb {B}$$ and, on the other hand, we find conditions under which every ball of radius smaller than 1 has a positive measure $$\Gamma$$.

## Keywords

Banach random walk Limit distribution Support of the measure Quasi-orthogonal Schauder basis

## Mathematics Subject Classification (2010)

60J15 60B12 60G42 60G46

## 1 Banach Random Walk in a Banach space

Construction of the Banach Random Walk in a Banach space was given in [3], so we present here only a brief description of this process.

Let $$\left( \mathbb {B},\left\| \cdot \right\| \right)$$ be an infinite-dimensional Banach space with a Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$ and let $$\{\pi _{n}, n\ge 0\}$$ be a sequence of projections $$\pi _{n}:\mathbb {B}\rightarrow \mathbb {B},$$ given by $$\pi _0 (x) \equiv 0\in \mathbb {B}$$ and $$\pi _{n}\left( x\right) =\sum _{k=1}^{n}x_{k}b_{k}$$ for $$x =\sum _{k=1}^{\infty }x_{k}b_{k}\in \mathbb {B}$$, $$n\ge 1$$. Denote
\begin{aligned} B=\left\{ x\in \mathbb {B}:\left\| x\right\| \le 1\right\} ,\quad B_{n}\left( 0,r\right) =\left\{ \pi _{n}\left( x\right) \in \mathbb {B}:\left\| \pi _{n}\left( x\right) \right\| \le r\right\} ,\quad n,r \ge 0 , \end{aligned}
and for $$\pi _{n-1}\left( x\right) \in B_{n-1}=B_{n-1}\left( 0,1\right)$$, where $$n\ge 1$$, put
\begin{aligned} \alpha _{n}= & {} \inf \left\{ t\in \mathbb {R}:\left\| \pi _{n-1}\left( x\right) +t b_{n}\right\| \le 1\right\} =\alpha _{n}(\pi _{n-1}\left( x\right) ),\\ \beta _{n}= & {} \sup \left\{ t\in \mathbb {R}:\left\| \pi _{n-1}\left( x\right) +tb_{n}\right\| \le 1\right\} =\beta _{n}(\pi _{n-1}\left( x\right) ). \end{aligned}
Without loss of generality we assume that $$\left\| b_{1}\right\| =1$$, but we do not require that $$\left\| b_{n}\right\| =1$$ for all $$n\ge 2$$. Obviously $$\beta _{1}=-\alpha _{1}$$, and in addition $$\alpha _{1}=-1$$ and $$\beta _{1}=1$$ whenever $$\left\| b_{1}\right\| =1$$, but in general $$\beta _{n}\ne -\alpha _{n}$$ for $$n \ge 2$$. Therefore we introduce the following notion: the Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$ is called quasi-orthogonal, if $$\alpha _{n+1}=-\beta _{n+1}$$ for all $$n\ge 1$$. Under the above assumption $$\left[ \alpha _{n},\beta _{n}\right]$$, $$n\ge 1$$, are bounded intervals in $$\mathbb {R}$$ with center zero, but in some situations they are reduced to the single point $$[0,0]=\{0\}$$.
Let $$\{G_{n}, n\ge 1\}$$ be arbitrary probability distributions satisfying condition $$G_{n}\left( \left[ -1,1\right] \right) =1$$ for all $$n\ge 1$$. Define inductively on a probability space $$\left( \Omega , \mathcal {F},P\right)$$ a sequence of dependent real-valued r.v.’s $$\left\{ X_{n},n\ge 1\right\}$$ and, associated with $$\left\{ X_{n},n\ge 1\right\}$$, $$\mathbb {B}$$-valued random elements (r.e.’s) $$\left\{ Z_{n},n\ge 1\right\}$$ as follows: let $$X_{1}$$ be a r.v. with distribution $$G_{1}$$ and let $$Z_{1}=X_{1}b_{1};$$ then $$X_{1}\left( \omega \right) \in \left[ \alpha _{1},\beta _{1}\right] =\left[ -1,1\right]$$, i.e., $$Z_{1}(\omega ) \in B_1$$ a.s., and thus we evaluate $$\beta _2 (Z_{1}(\omega ))$$, define $$X_{2}$$ as a r.v. distributed according to the scaled probability measure
\begin{aligned} G_{2}\left( \cdot / \beta _{2}\left( X_{1}\left( \omega \right) b_{1}\right) \right) = G_{2}\left( \cdot / \beta _{2}\left( Z_{1}\left( \omega \right) \right) \right) , \end{aligned}
whenever $$\beta _{2}\left( Z_{1}\left( \omega \right) \right) >0$$, and put $$Z_{2}=X_{1}b_{1}+X_{2}b_{2}$$. More generally, if r.v.’s $$X_{1},\ldots ,X_{n-1}$$ 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
\begin{aligned} G_{n}\left( \cdot / \beta _{n}\left( X_{1}\left( \omega \right) b_{1} + \cdots + X_{n-1}\left( \omega \right) b_{n-1} \right) \right) = G_{n}\left( \cdot / \beta _{n}\left( Z_{n-1}\left( \omega \right) \right) \right) , \end{aligned}
provided $$\beta _{n}\left( Z_{n-1}\left( \omega \right) \right) >0$$, and $$Z_{n}=X_{1}b_{1}+X_{2}b_{2}+\cdots +X_{n}b_{n}.$$ As was already mentioned, it may happen that for some $$n\ge 1$$ and $$Z_{n}\left( \omega \right) \in B_{n}$$ the interval $$\left[ \alpha _{n+1},\beta _{n+1}\right] = \left[ \alpha _{n+1}\left( Z_{n}\left( \omega \right) \right) ,\beta _{n+1}\left( Z_{n}\left( \omega \right) \right) \right]$$ reduces to the one-point set $$\left\{ 0\right\}$$; in such a case we assume that the measure $$G_{n+1}$$ is transformed so that it assigns the unit mass to the single point 0. Then $$Z_{n+1}(\omega ) = Z_n (\omega )$$, but 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 $$\{0\}$$, and thus the process is still continued.

According to the definition introduced in [3] the sequence of $$\mathbb {B}$$-valued r.e.’s $$\left\{ Z_{n},n\ge 1\right\}$$ obtained in this way is called Banach Random Walk (BRW) in the Banach space $$\mathbb {B}$$.

Construction of the Banach Random Walk in an infinite-dimensional separable Hilbert space $$\mathbb {H}$$ was motivated by Banach’s paper [1], where the so-called $$\mathfrak {L}$$-integral (i.e., integral of Lebesgue type) in abstract spaces was described. Namely, Banek [2] observed that a purely deterministic Banach’s [1] construction of the $$\mathfrak {L}$$-integral in $$\mathbb {H}$$ is closely related to the asymptotic properties of the Banach Random Walk in $$\mathbb {H}$$, and in fact the mentioned integral is equal to the limit of expectations of certain functionals acting on the Banach Random Walk. The main idea of Banach’s [1] approach which led to the definition of his $$\mathfrak {L}$$-integral was the symmetry of mappings as well as the symmetry of considered measures in $$\mathbb {R}^{n}$$, $$n\ge 1$$, and such a concept together with the Hahn–Banach theorem enabled him to prove the existence of the $$\mathfrak {L}$$-integral functional. Thus it is natural to demand that probability distributions $$G_{n}$$, $$n\ge 1$$, are symmetric in the sense that $$G\left( -A\right) =G\left( A\right)$$ for all $$A\in \mathcal {B}\left( \mathbb {R}\right)$$.

It was shown in [3] that under this assumption concerning distributions $$\{G_{n}, n\ge 1\}$$, the Banach Random Walk in a Banach space $$\mathbb {B}$$ is a martingale with respect to the natural filtration $$\{\mathcal {F}_n = \sigma (X_1,X_2,\ldots ,X_n), n\ge 1\}$$ (and in fact it is also a time-inhomogeneous Markov chain). Moreover, if the Banach space $$\mathbb {B}$$ in question possesses the Radon–Nikodym Property (RNP), cf. [4, 9], or [10] for the definition of this notion, then the process $$\{Z_n,n\ge 1\}$$ converges strongly a.s. in $$\mathbb {B}$$ and in $$L^{p}(\mathbb {B})$$ for all $$1\le p < \infty$$ to a r.e. $$\xi$$. The details of these considerations can be found in [3], thus we omit them here.

The aim of this paper is to describe the main properties of the limit distribution $$\Gamma = P\circ \xi ^{-1}$$ of the BRW $$\{Z_n ,n\ge 1\}$$ in a Banach space $$\mathbb {B}$$; in particular, we are interested in the description of the support $${\mathrm{supp\,}}\Gamma$$. It should be pointed out that for a class of bounded, Borel measurable functions $$\Phi$$ on the unit ball $$B\subset \mathbb {B}$$, the Banach–Lebesgue $$\mathfrak {L}$$-integral can be expressed as the expected value $$E\Phi (\xi )$$, see [3], thus the support of $$\xi$$ is of the significant importance, for it informs what the minimal domain of the integrand $$\Phi$$ should be.

## 2 Properties of Limit Distribution of the Banach Random Walk in a Banach Space

Throughout this section we assume that $$\mathbb {B}$$ is a Banach space which has the RNP and a quasi-orthogonal Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$, and $$\left\{ Z_{n},n\ge 1\right\}$$ is the BRW in $$\mathbb {B}$$ generated by a sequence of symmetric probability distributions $$\left\{ G_{n},n\ge 1\right\}$$ concentrated on the interval $$\left[ -1,1\right] \subset \mathbb {R}$$. Moreover, let $$\xi$$ denote the a.s. limit of the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ in $$\mathbb {B}$$, and let $$\Gamma =P\circ \xi ^{-1}$$ be the measure on the ball $$B=\left\{ x\in \mathbb {B}:\left\| x\right\| \le 1\right\}$$ induced by $$\xi$$.

Analyzing the construction of the process $$\left\{ Z_{n},n\ge 1\right\}$$ in a Banach space one may expect that the limit distribution $$\Gamma = P\circ \xi ^{-1}$$ of the BRW is concentrated on the surface $$S(0,1)=\left\{ x\in \mathbb {B} :\left\| x\right\| =1\right\}$$ of the closed unit ball $$B=\left\{ x\in \mathbb {B}:\left\| x\right\| \le 1\right\}$$. Obviously such a statement is heavily dependent on distributions $$\left\{ G_{n},n\ge 1\right\}$$, which exert an influence on r.v.’s $$\left\{ X_{n},n\ge 1\right\}$$, and in general need not be true. However, in the most interesting situation when $$\left\{ X_{n},n\ge 1\right\}$$ is a sequence of r.v.’s generated by identical distributions with support equal to the interval $$\left[ -1,1\right] \subset \mathbb {R}$$, this indeed is the case. To examine this problem we consider the BRW in a Banach space $$\mathbb { B}$$ satisfying all the above requirements. First we prove an auxiliary result.

### Lemma 1

For every $$x\in \mathbb {B}$$ such that $$\left\| \pi _{n-1}\left( x\right) \right\| \le r_{0}\le 1$$, the mapping
\begin{aligned} \left[ r_{0},\infty \right) \ni r\mapsto \beta _{n}\left( \pi _{n-1}\left( x\right) /r\right) ,\quad r_{0} > 0 , \end{aligned}
is a nondecreasing concave function. In consequence, it is continuous in the open interval $$\left( r_{0},\infty \right)$$, and a.e. right-hand side and left-hand side differentiable.

### Proof

Recall that $$\beta _{n}\left( \pi _{n-1}\left( x\right) \right)$$ is defined for $$\left\| \pi _{n-1}\left( x\right) \right\| \le 1$$ in such a way that $$\left\| \pi _{n-1}\left( x\right) +\beta _{n}\left( \pi _{n-1}\left( x\right) \right) b_{n}\right\| =1$$. Thus, if $$\left\| \pi _{n-1}\left( x\right) \right\| =r_{0}\le 1,$$ then $$\left\| \pi _{n-1}\left( x\right) /r+\beta _{n}\left( \pi _{n-1}\left( x\right) /r\right) b_{n}\right\| =1$$ for each $$r_{0}\le r<\infty$$. Since the unit ball is convex, for all $$r_{0}\le r_{1}\ne r_{2}<\infty$$ and $$\lambda _{1},\lambda _{2}\in [0,1]$$ such that $$\lambda _{1} + \lambda _{2}=1$$, we have
\begin{aligned} \left\| \lambda _{1}\frac{\pi _{n-1}\left( x\right) }{r_1}+\lambda _{2}\frac{\pi _{n-1}\left( x\right) }{r_2}+ \left[ \lambda _{1} \beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r_1}\right) + \lambda _{2}\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r_2}\right) \right] b_{n}\right\| \le 1. \end{aligned}
Hence and from the definition of $$\beta _{n}(\cdot )$$ it follows that
\begin{aligned} \lambda _{1} \beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r_1}\right) + \lambda _{2}\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r_2}\right) \le \beta _{n}\left( \lambda _{1}\frac{\pi _{n-1}\left( x\right) }{r_1}+ \lambda _{2}\frac{\pi _{n-1}\left( x\right) }{r_2}\right) , \end{aligned}
i.e., $$\left[ r_{0},\infty \right) \ni r\mapsto \beta _{n}\left( \pi _{n-1}\left( x\right) /r\right)$$ is a concave function. Consequently, it is continuous in the open interval $$\left( r_{0},\infty \right)$$, and a.e. right-hand side and left-hand side differentiable, cf. [5], Ch. V, Sect. 8, Th. 2.
Obviously, $$\pi _{n-1}\left( x\right) /r\rightarrow 0$$, $$r\rightarrow \infty$$, therefore $$\beta _{n}\left( \pi _{n-1}\left( x\right) /r\right) \rightarrow 1/\left\| b_{n}\right\|$$ as $$r\rightarrow \infty$$. Moreover, $$0\le \beta _{n}\left( \pi _{n-1}\left( x\right) /r\right) \le 1/\left\| b_{n}\right\|$$ for all $$r\in \left[ r_{0},\infty \right)$$; otherwise, in case when $$\beta _{n}\left( \pi _{n-1}\left( x\right) /r\right) >1/\left\| b_{n}\right\|$$ for some $$r\ge r_{0}$$, we would have
\begin{aligned}&\left\| \frac{\pi _{n-1}\left( x\right) }{r}+\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}-\frac{\pi _{n-1}\left( x\right) }{r} -\alpha _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \\&\quad =2\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) \cdot \left\| b_{n}\right\| >2\cdot \frac{1}{\left\| b_{n}\right\| } \cdot \left\| b_{n}\right\| =2, \end{aligned}
\begin{aligned} \left\| \frac{\pi _{n-1}\left( x\right) }{r}+\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \le 1, \quad \left\| \frac{\pi _{n-1}\left( x\right) }{r} +\alpha _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \le 1 . \end{aligned}
Hence it follows that $$\beta _{n}\left( \pi _{n-1}\left( x\right) /r\right)$$ is nondecreasing as $$r_{0} \le r \nearrow \infty$$. $$\square$$

To formulate the next result, some explanations are needed. The Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$ in a Banach space is called monotone, if for every choice of scalars $$\left\{ x_{n},n\ge 1\right\}$$ the sequence of real numbers $$\left\{ \left\| \sum \nolimits _{k=1}^{n}x_{k}b_{k}\right\| ,n\ge 1\right\}$$ is nondecreasing. It is fairly well known that for each Banach space with a Schauder basis there exists a norm equivalent to the original one, such that a given basis $$\left\{ b_{n},n\ge 1\right\}$$ in this space equipped with the new norm is monotone, see [6], Part I, Ch. I, p. 2. Thus, to avoid additional complications with a new norm concerning notation, in what follows we assume that the basis $$\left\{ b_{n},n\ge 1\right\}$$ in $$(\mathbb {B},\Vert \cdot \Vert )$$ is just monotone.

It is worth mentioning that many typical Schauder bases, such as the sequence of unit vectors in $$c_{0}$$ and $$\ell ^{p}$$ for $$1\le p < \infty$$, or the system of Haar functions in $$L^{p}[0,1]$$ for $$1\le p < \infty$$ are monotone; furthermore, to obtain this effect the usual norms of these spaces need not be changed, see, e.g., [6], Part I, Ch. I, p. 3.

### Theorem 1

Suppose that
\begin{aligned} \lim _{n\rightarrow \infty }\prod \limits _{k=1}^{n}G_{k}\left( \left[ -r,r \right] \right) =0 \end{aligned}
(1)
for some $$0<r<1$$. Then for the closed ball $$B\left( 0,r\right) =\left\{ x\in \mathbb {B}:\left\| x\right\| \le r\right\} ,$$ where $$0<r<1$$ is a fixed number, we have
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right) =0. \end{aligned}
In consequence, if condition (1) is satisfied for all $$0<r<1,$$ then the whole mass of the measure $$\Gamma =P\circ \xi ^{-1}$$ is concentrated on the unit sphere $$S\left( 0,1\right) =\left\{ x\in \mathbb {B}:\left\| x\right\| =1\right\} ,$$ so that $${\mathrm{supp\,}}\Gamma \subseteq S\left( 0,1\right) .$$

### Proof

Recall that to define the first n steps of the BRW in a Banach space $$\mathbb {B}$$ with a quasi-orthogonal Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$ we have to use the following transformation $$\Theta _{n}: K_n^0 (0,1) \rightarrow (-1,1)^n \subset \mathbb {R}^{n},$$
\begin{aligned} y_{1}= & {} x_{1} , \nonumber \\ y_{2}= & {} \frac{x_{2}}{\beta _{2}\left( x_{1}b_{1}\right) }\, , \nonumber \\ y_{3}= & {} \frac{x_{3}}{\beta _{3}\left( x_{1}b_{1}+x_{2}b_{2}\right) }\, , \nonumber \\&\vdots \nonumber \\ y_{n}= & {} \frac{x_{n}}{\beta _{n}\left( x_{1}b_{1}+\cdots +x_{n-1}b_{n-1}\right) }\, , \end{aligned}
(2)
where $$K_{n}\left( 0,r\right) =\left\{ \left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:\left\| x_{1}b_{1}+\cdots +x_{n}b_{n}\right\| \le r\right\}$$, and $$K_{n}^{0}\left( 0,r\right) =\, \text {Int}\, K_{n}\left( 0,r\right) =\left\{ \left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:\left\| x_{1}b_{1}+\cdots +x_{n}b_{n}\right\| < r\right\}$$, $$0<r<\infty$$, $$n\ge 1$$. Notice that if $$\left( x_1 ,\ldots ,x_{k-1},0,\ldots ,0\right) \in K_n^0 (0,1)$$ for some $$1< k\le n$$, then there exists an open ball with center at this point contained in $$K_n^0 (0,1)$$, thus $$\beta _{k}\left( x_{1}b_{1}+\cdots +x_{k-1}b_{k-1}\right) >0$$ and so $$\Theta _n$$ is well defined.
To find the inverse transformation $$T_{n} = \Theta _n^{-1}$$ to (2) we introduce recursively a sequence of mappings: $$A_{1}\equiv 1,$$$$A_{2}\left( y_{1}\right) =\beta _{2}\left( y_{1}A_{1}b_{1}\right) =\beta _{2}\left( y_{1}b_{1}\right) ,$$$$A_{3}\left( y_{1},y_{2}\right) =\beta _{3}\left( y_{1}A_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}\right) =\beta _{3}\left( y_{1}b_{1}+y_{2}\beta _{2}\left( y_{1}b_{1}\right) b_{2}\right) ,\ldots$$
\begin{aligned} A_{n}\left( y_{1},y_{2},\ldots ,y_{n-1}\right)= & {} \beta _{n}\left( y_{1}A_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+y_{3}A_{3}\left( y_{1},y_{2}\right) b_{3}\right. \nonumber \\&\left. + \cdots +y_{n-1}A_{n-1}\left( y_{1},y_{2},\ldots ,y_{n-2}\right) b_{n-1}\right) . \end{aligned}
(3)
Then the transformation $$T_{n}: (-1,1)^n \rightarrow K_n^0 (0,1)$$ is given by
\begin{aligned} x_{1}= & {} y_{1}\cdot A_{1}=y_{1}, \nonumber \\ x_{2}= & {} y_{2}\cdot A_{2}\left( y_{1}\right) =y_{2}\cdot \beta _{2}\left( y_{1}b_{1}\right) , \nonumber \\ x_{3}= & {} y_{3}\cdot A_{3}\left( y_{1},y_{2}\right) =y_{3}\cdot \beta _{3}\left( y_{1}b_{1}+y_{2}\beta _{2}\left( y_{1}b_{1}\right) b_{2}\right) , \nonumber \\&\vdots \nonumber \\ x_{n}= & {} y_{n}\cdot A_{n}\left( y_{1},y_{2},\ldots ,y_{n-1}\right) . \end{aligned}
(4)
Equations (4) can be verified by induction on the basis of (2). As can be seen, $$\Theta _{n}\left( K_{n}^{0}\left( 0,1\right) \right) = \left( -1,1\right) ^{n}$$ along with $$T_{n}\left( \left( -1,1\right) ^{n} \right) =K_{n}^{0}\left( 0,1\right) ,$$ and both these mappings restricted to the domains considered here are one-to-one. The map $$T_n$$ is also well defined in the whole closed cube $$[-1,1]^n$$, but then in general it is not injective, in particular—on the boundary $$[-1,1]^n \setminus (-1,1)^n$$. Thus, although $$\Theta _n$$ is in fact the inverse mapping to $$\left. T_n \right| _{(-1,1)^n}$$, instead of the inverse transformation to $$T_n$$ acting on $$[-1,1]^n$$ which need not exist, we must investigate inverse images $$T_n^{-1} (B)$$ of Borel sets $$B\in \mathcal {B}\left( K_n(0,1) \right)$$.
Let $$(Y_1,\ldots ,Y_n)$$ be a random vector with values in $$[-1,1]^n$$ and distribution $$\prod _{k=1}^n G_k$$. Taking into account the construction of BRW, we conclude that $$(X_1,\ldots ,X_n) = T_n (Y_1,\ldots ,Y_n)$$. Observe that each map $$\beta _{k}\left( x_{1}b_{1}+\cdots +x_{k-1}b_{k-1}\right)$$ is a continuous function of $$\left( x_{1},\ldots ,x_{k-1}\right) \in K_{k-1} (0,1)$$; to see this, consider sets of the form $$p_{k-1}\left( S_{+}\cap (\mathbb {R}^{k-1}\times F) \right) = \left( \beta _k^{\prime }\right) ^{-1}(F)$$, where $$S_{+}$$ is the graph of $$\beta _k^{\prime }(x_1,\ldots ,x_{k-1})=\beta _k \left( x_{1}b_{1}+\cdots +x_{k-1}b_{k-1}\right)$$, $$p_{k-1}(x_1,\ldots ,x_k) = (x_1,\ldots ,x_{k-1})$$ is the usual projection of $$\mathbb {R}^k$$ onto $$\mathbb {R}^{k-1}$$, and F is a closed subset of $$\mathbb {R}$$. Since $$T_n$$ is a composition of continuous functions with $$\beta _k$$, we conclude that $$T_n$$ is continuous as well and in consequence $$(X_1,\ldots ,X_n)$$ is a random vector. The distribution of $$(X_1,\ldots ,X_n)$$ is equal
\begin{aligned} P\circ (X_1,\ldots ,X_n)^{-1} = P\circ (Y_1,\ldots ,Y_n)^{-1}\circ T_n^{-1} = \left( \prod _{k=1}^n G_k \right) \circ T_n ^{-1} . \end{aligned}
From (4) we infer that for a fixed $$0<r < 1$$,
\begin{aligned}&\left\| x_{1}b_{1}+x_{2}b_{2}+\cdots +x_{n}b_{n}\right\| \le r\nonumber \\&\quad \Leftrightarrow ~\left\| y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n} A_{n}\left( y_{1},\ldots ,y_{n-1}\right) b_{n}\right\| \le r. \end{aligned}
(5)
Define
\begin{aligned} D_{n}\left( r\right)= & {} T_{n}^{-1}\left( K_{n}(0,r)\right) =\left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n}:\left\| y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}\right. \right. \nonumber \\&\left. \left. +\cdots +y_{n} A_{n}\left( y_{1},\ldots ,y_{n-1}\right) b_{n}\right\| \le r\right\} , \end{aligned}
cf. (5). Since $$K_n(0,r)$$ is a closed subset of $$K_n (0,1)$$, the set $$D_n (r)$$ is a Borel subset of $$[-1,1]^n$$.
Divide both sides of (5) by r and observe that if $$\left( y_{1},\ldots ,y_{n}\right) \in D_{n}\left( r\right)$$, then by definition of $$\beta _{n}\left( \pi _{n-1}\left( x\right) \right)$$ we obtain
\begin{aligned} \left| \frac{y_{n}\cdot A_{n}\left( y_{1},\ldots ,y_{n-1}\right) }{r} \right| \le \beta _{n}\left( \frac{y_{1}b_{1}+\cdots +y_{n-1} A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}}{r}\right) , \end{aligned}
where $$A_{n}\left( y_{1},y_{2},\ldots ,y_{n-1}\right)$$ is given by (3 ), i.e.,
\begin{aligned} \left| y_{n}\right| \le \frac{r\cdot \beta _{n}\left( \displaystyle \frac{ y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n-1} A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}}{r}\right) }{\beta _{n}\left( y_{1}A_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n-1}A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}\right) }. \end{aligned}
(6)
Applying Lemma 1 we have
\begin{aligned}&\beta _{n}\left( \displaystyle \frac{ y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n-1} A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}}{r}\right) \nonumber \\&\quad \le \beta _{n}\left( y_{1}A_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n-1}A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}\right) , \end{aligned}
for $$r \le 1$$. Taking into account the above estimate and (6) we conclude that $$\left| y_{n}\right| \le r.$$ In consequence,
\begin{aligned} D_{n}\left( r\right) \subseteq \left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n}:\left| y_{n}\right| \le r\right\} . \end{aligned}
Moreover, since the basis $$\{ b_{n}, n\ge 1 \}$$ is monotone, condition (5) implies that
\begin{aligned} \left\| y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n-1} A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}\right\| \le r. \end{aligned}
In other words,
\begin{aligned} D_{n}\left( r\right)\subseteq & {} \left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n}: \left\| y_{1}b_{1}+\cdots +y_{n-1} A_{n-1}\left( y_{1},\ldots ,y_{n-2}\right) b_{n-1}\right\| \right. \nonumber \\&\left. \le r,\left| y_{n}\right| \le r\right\} . \end{aligned}
Arguing in a similar way as above we infer that $$\left| y_{n-1}\right| \le r,$$ next $$\left| y_{n-2}\right| \le r,$$ etc., and finally, from $$\left\| y_{1}b_{1}\right\| \le r$$ and $$\left\| b_{1}\right\| =1,$$ it follows that $$\left| y_{1}\right| \le r$$. Thus we conclude that
\begin{aligned} D_{n}\left( r\right) \subseteq \left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n}:\left| y_{1}\right| \le r,\ldots ,\left| y_{n}\right| \le r\right\} =\left[ -r,r\right] ^{n} , \end{aligned}
i.e., $$T_{n}^{-1}\left( K_{n}\left( 0,r\right) \right) = D_{n}\left( r\right) \subseteq \left[ -r,r\right] ^{n} .$$ Hence it follows that
\begin{aligned} \Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \right)= & {} P\circ \xi ^{-1}\left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \right) \nonumber \\= & {} P\left[ \, \pi _{n}\left( \xi \right) \in B_{n}\left( 0,r\right) \right] \nonumber \\= & {} P\left[ Z_{n}\in B_{n}\left( 0,r\right) \right] =P\left[ \left( X_{1},\ldots ,X_{n}\right) \in K_{n}\left( 0,r\right) \right] \nonumber \\= & {} P\left[ \, T_n \left( Y_{1},\ldots ,Y_{n}\right) \in K_{n}\left( 0,r\right) \right] \nonumber \\= & {} P\left[ \left( Y_{1},\ldots ,Y_{n}\right) \in T_{n}^{-1} \left( K_{n}\left( 0,r\right) \right) \right] \nonumber \\= & {} \left( \prod \limits _{k=1}^{n}G_{k}\right) \left( T_{n}^{-1}\left( K_{n}\left( 0,r\right) \right) \right) =\left( \prod \limits _{k=1}^{n}G_{k}\right) \left( D_{n}\left( r\right) \right) \nonumber \\\le & {} \prod \limits _{k=1}^{n}G_{k}\left( \left[ -r,r\right] \right) . \end{aligned}
(7)
In fact we have
\begin{aligned} \Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \right) =\Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) , \end{aligned}
(8)
as we already know that $${\mathrm{supp\,}}\Gamma \subseteq B .$$ Furthermore,
\begin{aligned} \pi _{1}^{-1}\left( B_{1}\left( 0,r\right) \right) \cap B \supseteq \pi _{2}^{-1}\left( B_{2}\left( 0,r\right) \right) \cap B \supseteq \cdots \supseteq \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \supseteq \cdots \end{aligned}
(9)
and
\begin{aligned} \bigcap \limits _{n=1}^{\infty }\pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B = B \left( 0,r\right) . \end{aligned}
(10)
Consequently,
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right)= & {} \Gamma \left( \bigcap \limits _{n=1}^{\infty }\pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) \\= & {} \lim _{n\rightarrow \infty }\Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) \le \lim _{n\rightarrow \infty }\prod \limits _{k=1}^{n}G_{k}\left( \left[ -r,r\right] \right) =0. \end{aligned}
If $$0<r<1$$ in (1) can be arbitrary, the final conclusion $$\Gamma \left( S\left( 0,1\right) \right) =1$$ of the theorem, which can be rewritten also in the form $${\mathrm{supp\,}}\Gamma \subseteq S\left( 0,1\right) ,$$ is evident. $$\square$$

### Corollary 1

If $$\left\{ G_{n},n\ge 1\right\}$$ is a sequence of identical distributions $$G_{n}=G,$$$$n\ge 1,$$ such that $$G\left( \left[ -r,r\right] \right) <1$$ for each $$0<r<1,$$ then the assertion of Theorem 1 remains valid. In particular, if $$G_{n}=U,$$$$n\ge 1,$$ are identical uniform distributions on $$\left[ -1,1\right] ,$$ then Theorem 1 holds true.

We are able to prove as well a result going in the opposite direction. To formulate the next theorem, given any $$0<r\le 1$$, we choose a sequence of positive real numbers $$\left\{ q_{n},n\ge 1\right\}$$ satisfying condition
\begin{aligned} 0<q_{n}<\left( \sqrt{r^{2}+4r}-r\right) /2 \le \left( \sqrt{5}-1\right) /2 ,\quad n \ge 1 , \end{aligned}
(i.e., $$q_{n}^{2}+q_{n}^{3}+q_{n}^{4}+\cdots =q_{n}^{2}/\left( 1-q_{n}\right) <r$$ ) and put
\begin{aligned} s_{k,n}=q_{n}^{k}+q_{n}^{k+1}+\cdots +q_{n}^{n}\quad \text {for }\quad 2\le k\le n ,\quad s_{n+1,n}=0 ,\quad n \ge 1 . \end{aligned}

### Theorem 2

Assume that for a given $$0<r\le 1,$$ there exists a sequence of numbers $$\left\{ q_{n},n\ge 1\right\} \subset \mathbb {R}$$ satisfying the above requirements, such that
\begin{aligned}&\limsup _{n\rightarrow \infty }G_{1}\left( \left[ -\frac{\left( r-s_{2,n}\right) }{\left( 1-s_{2,n}\right) },\frac{\left( r-s_{2,n}\right) }{ \left( 1-s_{2,n}\right) }\right] \right) \nonumber \\&\quad \cdot \prod \limits _{k=2}^{n}G_{k}\left( \left[ -\frac{q_{n}^{~k}}{\left( 1-s_{k+1,n}\right) },\frac{q_{n}^{~k}}{ \left( 1-s_{k+1,n}\right) }\right] \right) = c_{r}>0. \end{aligned}
(11)
Then we have
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right) \ge c_{r}>0, \end{aligned}
therefore if  $$0<r<1$$, then the whole mass of the measure $$\Gamma$$ cannot be concentrated on the unit sphere $$S\left( 0,1\right) =\left\{ x\in \mathbb {B}:\left\| x\right\| =1\right\} .$$

### Proof

Let $$\Theta _{n}$$ and $$T _{n}$$ be the transformations given by ( 2) and (4) resp. Notice that then
\begin{aligned}&\beta _{k}\left( y_{1}A_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{k-1}A_{k-1}\left( y_{1},y_{2},\ldots ,y_{k-2}\right) b_{k-1}\right) \\&\quad =A_{k}\left( y_{1},y_{2},\ldots ,y_{k-1}\right) ,\quad 2\le k\le n, \end{aligned}
cf. (3). Since for every fixed $$x,y\in \mathbb {B}$$ the mapping $$t\mapsto \left\| x+ty\right\|$$ is a continuous function of the parameter $$t\in \mathbb {R}$$, we have the following system of equivalent conditions:
\begin{aligned}&\left\| \left( 1-s_{2,n}\right) y_{1}b_{1}\right\| \le r-s_{2,n}\Leftrightarrow \left| y_{1}\right| \le \frac{r-s_{2,n}}{ 1-s_{2,n}}\, ,\nonumber \\&\left\| q_{n}^{2}y_{1}b_{1}+\left( 1-s_{3,n}\right) y_{2}A_{2}b_{2}\right\| \le q_{n}^{2}\Leftrightarrow \left| y_{2}\right| \le \frac{q_{n}^{2}}{1-s_{3,n}},\nonumber \\&\left\| q_{n}^{3}\left( y_{1}b_{1}+y_{2}A_{2}b_{2}\right) +\left( 1-s_{4,n}\right) y_{3}A_{3}b_{3}\right\| \le q_{n}^{3} \Leftrightarrow \left| y_{3}\right| \le \frac{q_{n}^{3}}{ 1-s_{4,n}}, \nonumber \\&\quad \qquad \qquad \qquad \vdots \nonumber \\&\left\| q_{n}^{n}\left( y_{1}b_{1}+y_{2}A_{2}b_{2}+\cdots +y_{n-1}A_{n-1}b_{n-1}\right) +y_{n}A_{n}b_{n}\right\| \le q_{n}^{n}\Leftrightarrow \left| y_{n}\right| \le q_{n}^{n} \end{aligned}
(12)
(to simplify the writing, we put here $$A_{k}=A_{k}\left( y_{1},y_{2},\ldots ,y_{k-1}\right)$$, $$2\le k\le n$$). Summing all the inequalities on the left-hand side of (12) we conclude that
\begin{aligned}&\left\| y_{1}b_{1}+y_{2}A_{2}\left( y_{1}\right) b_{2}+\cdots +y_{n} A_{n}\left( y_{1},\ldots ,y_{n-1}\right) b_{n}\right\| \le \left\| \left( 1-s_{2,n}\right) y_{1}b_{1}\right\| \nonumber \\&\qquad + \left\| q_{n}^{2}y_{1}b_{1}+\left( 1-s_{3,n}\right) y_{2}A_{2}b_{2}\right\| +\left\| q_{n}^{3}\left( y_{1}b_{1}+y_{2}A_{2}b_{2}\right) +\left( 1-s_{4,n}\right) y_{3}A_{3}b_{3}\right\| \nonumber \\&\qquad +\cdots +\left\| q_{n}^{n}\left( y_{1}b_{1}+y_{2}A_{2}b_{2}+\cdots +y_{n-1}A_{n-1}b_{n-1}\right) +y_{n}A_{n}b_{n}\right\| \nonumber \\&\quad \le r-s_{2,n}+q_{n}^{2}+q_{n}^{3}+\cdots +q_{n}^{n}=r, \end{aligned}
thus
\begin{aligned}&\Delta _{n}\left( r,q_{n}\right) \\&\quad :=\left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n}:\left| y_{1}\right| \le \frac{r-s_{2,n}}{1-s_{2,n}} ,\left| y_{2}\right| \le \frac{q_{n}^{2}}{1-s_{3,n}},\ldots ,\left| y_{n}\right| \le q_{n}^{n}\right\} \\&\quad \subseteq \left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n} : \left\| y_{1}b_{1}+\cdots +y_{n} A_{n}\left( y_{1},\ldots ,y_{n-1}\right) b_{n}\right\| \le r\right\} =D_{n}\left( r\right) . \end{aligned}
Hence, by analogy to (7)–(8), it follows that
\begin{aligned}&\Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) =\left( \prod \limits _{k=1}^{n}G_{k}\right) \left( D_{n}\left( r\right) \right) \ge \left( \prod \limits _{k=1}^{n}G_{k}\right) \left( \Delta _{n}\left( r,q_{n}\right) \right) \\&\quad =G_{1}\left( \left[ -\frac{r-s_{2,n}}{1-s_{2,n}},\frac{r-s_{2,n}}{1-s_{2,n}} \right] \right) \prod \limits _{k=2}^{n}G_{k}\left( \left[ -\frac{ q_{n} ^{~k}}{1-s_{k+1,n}},\frac{ q_{n} ^{~k}}{1-s_{k+1,n}} \right] \right) . \end{aligned}
Passing to the limit as $$n\rightarrow \infty$$, on account of (9 )–(10) and the assumption (11) we finally conclude that $$\Gamma \left( B\left( 0,r\right) \right) \ge c_{r}>0$$. $$\square$$

Combining Theorems 1 and 2 we obtain the following result.

### Corollary 2

Let $$\left\{ G_{n},n\ge 1\right\}$$ be a sequence of probability distributions concentrated on the interval $$\left[ -1,1\right] \subset \mathbb {R}$$ such that condition (1) is satisfied for all r, $$0<r<r_{1}<1$$, and there exists a sequence of positive numbers $$\left\{ q_{n},n\ge 1\right\} \subset \mathbb {R}$$ such that $$q_{n}^{2}+q_{n}^{3}+q_{n}^{4}+\cdots =q_{n}^{2}/\left( 1-q_{n}\right) <r_{1}$$, $$n \ge 1$$, along with condition (11) satisfied for $$r=r_{1}.$$ Then
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right) =0,\quad 0<r<r_{1},\quad \text {and}\quad \Gamma \left( B\left( 0,r_{1}\right) \right) \ge c_{r_{1}}>0. \end{aligned}
Thus $${\mathrm{supp\,}}\Gamma \subseteq B \setminus B^{0}\left( 0,r_{1}\right) ,$$ where $$B^{0}\left( 0,r_{1}\right) =\left\{ x\in \mathbb {B}:\left\| x\right\| <r_{1}\right\} .$$

### Remark 1

It is obvious that if $$c_{r}=1$$ for some $$0<r<1$$ in condition (11), then $$\Gamma \left( B(0,r)\right) = 1$$, thus in such a case $${\mathrm{supp\,}}\Gamma \subseteq B(0,r)$$.

## 3 Limit Distribution of the Banach Random Walk in $$\ell ^{p}$$

The assertion of Theorem 1 is quite clear and undoubtedly the assumptions of this result can be satisfied, but it is not so evident that there can be found a sequence of numbers $$\left\{ q_{n},n\ge 1\right\}$$ satisfying conditions specified in Theorem or Corollary 2. Therefore to solve the problem, we consider in more detail the space $$\mathbb {B}=\ell ^{p},$$ i.e., the separable Banach space of all infinite sequences $$x=\left( x_{1},x_{2},\ldots \right) \subset \mathbb {R}$$ with norm $$\left| x\right| _{p}=\left( \sum \nolimits _{n=1}^{\infty }\left| x_{n}\right| ^{p}\right) ^{1/p}<\infty$$, $$1\le p < \infty$$. As will be seen later, in such a case not merely a fixed ball $$B\left( 0,r\right) \subset \ell ^{p}$$ has a positive measure $$\Gamma$$ for suitably chosen distributions $$\left\{ G_{n},n\ge 1\right\}$$, but even for all $$0<r<1$$ we may have $$\Gamma \left( B\left( 0,r\right) \right) >0.$$

### Proposition 1

Let $$\left\{ Z_{n},n\ge 1\right\}$$ be the BRW in $$\ell ^{p}$$, $$1\le p < \infty$$, generated by a sequence $$\left\{ G_{n},n\ge 1\right\}$$ of symmetric probability distributions on the interval $$\left[ -1,1\right]$$, let  $$\xi$$ be the a.s. limit of the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ in $$\ell ^{p},$$ and let $$\Gamma =P\circ \xi ^{-1}$$ denote the measure on $$B=\left\{ x\in \ell ^{p}:\left| x\right| _{p} \le 1\right\}$$ induced by $$\xi$$. Consider a triangular array $$\left\{ c_{k,n},1\le k\le n,n\ge 1\right\}$$ of real numbers satisfying the following conditions:
\begin{aligned} 0<c_{k,n}<1\quad \text {for all}\quad k,n,\quad \text {and}\quad \sum \limits _{k=1}^{n}c_{k,n}=1,\quad n=1,2,\ldots \end{aligned}
Assume that the distributions $$G_{n},$$$$n\ge 1$$, are chosen in such a way that
\begin{aligned} \limsup \limits _{n\rightarrow \infty }\prod \limits _{k=1}^{n}G_{k}\left( \left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}\right] \right) =c_{r}>0 \end{aligned}
for a fixed $$0<r<1$$. Then for the closed ball $$B\left( 0,r\right) =\left\{ x\in \ell ^{p}:\left| x\right| _p \le r\right\} ,$$ where $$0<r<1,$$ we have
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right) \ge c_{r}>0. \end{aligned}
Consequently, in such a case the whole mass of measure $$\Gamma$$ is not concentrated on the unit sphere $$S\left( 0,1\right) =\left\{ x\in \ell ^{p}:\left| x\right| _p =1\right\} .$$

### Proof

As in the proof of Theorem 1, we now consider two transformations: $$\Theta _{n}:K_{n}^{0}\left( 0,1\right) =\left\{ \left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:\left| x_{1}\right| ^{p}+\cdots +\left| x_{n}\right| ^{p} < 1\right\} \rightarrow \left( -1,1\right) ^{n}$$ and $$T _{n}:\left( -1,1\right) ^{n} \rightarrow K_{n}^{0}\left( 0,1\right)$$, given by
\begin{aligned} y_{1}= & {} x_{1}, \nonumber \\ y_{2}= & {} \frac{x_{2}}{\left( 1-\left| x_{1}\right| ^{p}\right) ^{1/p}}, \nonumber \\ y_{3}= & {} \frac{x_{3}}{\left[ 1-\left( \left| x_{1}\right| ^{p}+\left| x_{2}\right| ^{p}\right) \right] ^{1/p}}, \nonumber \\&\vdots \nonumber \\ y_{n}= & {} \frac{x_{n}}{\left[ 1-\left( \left| x_{1}\right| ^{p}+\cdots +\left| x_{n-1}\right| ^{p}\right) \right] ^{1/p}}, \end{aligned}
(13)
and
\begin{aligned} x_{1}= & {} y_{1}, \nonumber \\ x_{2}= & {} y_{2}\cdot \left( 1-\left| y_{1}\right| ^{p}\right) ^{1/p}, \nonumber \\ x_{3}= & {} y_{3}\cdot \left[ \left( 1-\left| y_{1}\right| ^{p}\right) \cdot \left( 1-\left| y_{2}\right| ^{p}\right) \right] ^{1/p}, \nonumber \\&\qquad \vdots \nonumber \\ x_{n}= & {} y_{n}\cdot \left[ \left( 1-\left| y_{1}\right| ^{p}\right) \cdot \ldots \cdot \left( 1-\left| y_{n-1}\right| ^{p}\right) \right] ^{1/p}, \end{aligned}
(14)
resp. To derive (14), proceed by induction. We may also extend $$T_n$$ to the whole closed cube $$[-1,1]^n$$ by (14). Then $$P\circ (X_1,\ldots ,X_n)^{-1} = \left( \prod _{k=1}^{n} G_k \right) \circ T_n ^{-1}$$, as well as $$\Theta _n ^{-1} = \left. T_n \right| _{(-1,1)^n}$$ is the inverse map to $$\Theta _n$$. Notice next that
\begin{aligned} \left| x_{1}\right| ^{p}+\left| x_{2}\right| ^{p}+\cdots +\left| x_{n}\right| ^{p}=1-\left( 1-\left| y_{1}\right| ^{p}\right) \cdot \left( 1-\left| y_{2}\right| ^{p}\right) \cdot \ldots \cdot \left( 1-\left| y_{n}\right| ^{p}\right) , \end{aligned}
thus for a fixed $$0<r<1$$ we have
\begin{aligned}&\left| x_{1}\right| ^{p}+\left| x_{2}\right| ^{p}+\cdots +\left| x_{n}\right| ^{p}\le r^{p}\nonumber \\&\quad \Leftrightarrow ~\left( 1-\left| y_{1}\right| ^{p}\right) \cdot \left( 1-\left| y_{2}\right| ^{p}\right) \cdot \ldots \cdot \left( 1-\left| y_{n}\right| ^{p}\right) \ge 1-r^{p}. \end{aligned}
(15)
Arguing similarly as above we observe that
\begin{aligned}&\left( \bigwedge \limits _{1\le k\le n}~\left( 1-\left| y_{k}\right| ^{p}\right) \ge \left( 1-r^{p}\right) ^{c_{k,n}}\right) \\&\quad \Rightarrow ~\left( 1-\left| y_{n}\right| ^{p}\right) \left( 1-\left| y_{n}\right| ^{p}\right) \cdot \ldots \cdot \left( 1-\left| y_{n}\right| ^{p}\right) \ge \left( 1-r^{p}\right) ^{\sum \nolimits _{k=1}^{n}c_{k,n}}=\left( 1-r^{p}\right) . \end{aligned}
Moreover, for each fixed k
\begin{aligned} \left( 1-\left| y_{k}\right| ^{p}\right) \ge \left( 1-r^{p}\right) ^{c_{k,n}}\quad \Leftrightarrow \quad \left| y_{k}\right| \le \left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}. \end{aligned}
Hence
\begin{aligned}&\left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{1,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{1,n}}\right] ^{1/p}\right] \\&\quad \times \cdots \times \left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{n,n}}\right] ^{1/p}, \left[ 1-\left( 1-r^{p}\right) ^{c_{n,n}}\right] ^{1/p}\right] \subset D_{n}\left( r\right) , \end{aligned}
where
\begin{aligned} D_{n}\left( r\right) = T_n ^{-1} \left( K_n (0,r) \right)= & {} \left\{ \left( y_{1},\ldots ,y_{n}\right) \in \left[ -1,1\right] ^{n} :\left( 1-\left| y_{1}\right| ^{p}\right) \left( 1-\left| y_{2}\right| ^{p}\right) \right. \\&\left. \cdot \ldots \cdot \left( 1-\left| y_{n}\right| ^{p}\right) \ge 1-r^{p}\right\} . \end{aligned}
Therefore, for each $$n\ge 1$$ we have
\begin{aligned}&\prod \limits _{k=1}^{n}G_{k}\left( \left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}\right] \right) \\&\quad \le \left( G_{1}\times G_{2}\times \cdots \times G_{n}\right) \left( D_{n}\left( r\right) \right) =\Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) , \end{aligned}
cf. (7)–(8). Referring to (9)–(10) we obtain
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right)= & {} \Gamma \left( \bigcap \limits _{n=1}^{\infty }\pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) = \lim _{n\rightarrow \infty }\Gamma \left( \pi _{n}^{-1}\left( B_{n}\left( 0,r\right) \right) \cap B \right) \\\ge & {} \limsup _{n\rightarrow \infty }\prod \limits _{k=1}^{n}G_{k}\left( \left[ - \left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}\right] \right) \\= & {} c_{r}>0, \end{aligned}
which concludes the proof. $$\square$$

The example presented below shows that the distribution of the limit random element $$\xi$$ of the BRW in the Banach space $$\mathbb {B}=\ell ^{p}$$ may in some sense be split uniformly on balls centered at 0.

### Example 1

Let $$G_{k},\,k\ge 1,$$ be symmetric probability distributions on $$\left[ -1,1\right]$$ such that
\begin{aligned} G_{k}\left( \left[ -z,z\right] \right) =\left\{ 1-\left( 1-z^{p}\right) ^{2^{k}}\right\} ^{1/p2^{k}}\quad \text {for}\quad 0\le z\le 1,~ k\ge 1. \end{aligned}
(16)
Notice that
\begin{aligned} G_{k}\left( \left[ -z,z\right] \right) \rightarrow 0\text { as } z\rightarrow 0,\quad G_{k}\left( \left[ -z,z\right] \right) \rightarrow 1\text { as }z\rightarrow 1, \end{aligned}
and since
\begin{aligned} \left\{ G_{k}\left( \left[ -z,z\right] \right) ^{p2^{k}}\right\} ^{\prime }=-2^{k}\left( 1-z^{p}\right) ^{2^{k}-1}\left( -pz^{p-1}\right) =2^{k}pz^{p-1}\left( 1-z^{p}\right) ^{2^{k}-1}>0 \end{aligned}
for $$0<z<1,$$ it follows that the maps $$G_{k}\left( \left[ -z,z\right] \right)$$ are increasing in the interval $$0<z<1.$$ Therefore $$G_{k},$$$$k\ge 1,$$ are well defined. Consider the triangular array $$\left\{ c_{k,n},1\le k\le n,n\ge 1\right\}$$ of real numbers given by
\begin{aligned} c_{k,n}=1/2^{k}\quad \text {for}\quad 1\le k\le n-1,\quad \text {and}\quad c_{n,n}=1/2^{n-1}. \end{aligned}
Clearly, we have
\begin{aligned} \sum \limits _{k=1}^{n}c_{k,n}=\sum \limits _{k=1}^{n-1}\frac{1}{2^{k}}+\frac{1}{ 2^{n-1}}= \frac{1}{2}\cdot \frac{1-1/2^{n-1}}{1-1/2} +\frac{1}{2^{n-1}}=1. \end{aligned}
Substituting $$z=\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}$$ in the definition of $$G_{k}\left( \left[ -z,z\right] \right)$$ we obtain
\begin{aligned} \left\{ 1-\left( 1-z^{p}\right) ^{2^{k}}\right\} ^{1/p2^{k}}= & {} \left\{ 1-\left( 1-\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{p\cdot 1/p}\right) ^{2^{k}}\right\} ^{1/p2^{k}}\\= & {} \left\{ 1-\left( 1-r^{p}\right) ^{c_{k,n}\cdot 2^{k}}\right\} ^{1/p2^{k}}=\left\{ r^{p}\right\} ^{1/p2^{k}}=r^{1/2^{k}} \end{aligned}
for $$1\le k\le n-1 ,$$ and
\begin{aligned} \left\{ 1-\left( 1-z^{p}\right) ^{2^{n}}\right\} ^{1/p2^{n}}= & {} \left\{ 1-\left( 1-\left[ 1-\left( 1-r^{p}\right) ^{c_{n,n}}\right] ^{p\cdot 1/p}\right) ^{2^{n}}\right\} ^{1/p2^{n}}\\= & {} \left\{ 1-\left( 1-r^{p}\right) ^{c_{n,n}\cdot 2^{n}}\right\} ^{1/p2^{n}}=\left\{ 1-\left( 1-r^{p}\right) ^{2}\right\} ^{1/p2^{n}}\\= & {} r^{1/2^{n}}\cdot \left( 2-r^{p}\right) ^{1/p2^{n}} \end{aligned}
for $$k=n .$$ Hence
\begin{aligned}&\prod \limits _{k=1}^{n}G_{k}\left( \left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}\right] \right) \\&\quad =\left( \prod \limits _{k=1}^{n-1}r^{1/2^{k}}\right) \cdot r^{1/2^{n}}\cdot \left( 2-r^{p}\right) ^{1/p2^{n}}=r^{\sum \nolimits _{k=1}^{n-1}\left( 1/2^{k}\right) }\cdot r^{1/2^{n}}\cdot \left( 2-r^{p}\right) ^{1/p2^{n}}\\&\quad =r^{1-1/2^{n-1}+1/2^{n}}\cdot \left( 2-r^{p}\right) ^{1/p2^{n}}=r^{1-1/2^{n}}\cdot \left( 2-r^{p}\right) ^{1/p2^{n}}\rightarrow r, \end{aligned}
so that
\begin{aligned} \limsup \limits _{n\rightarrow \infty }\prod \limits _{k=1}^{n}G_{k}\left( \left[ -\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p},\left[ 1-\left( 1-r^{p}\right) ^{c_{k,n}}\right] ^{1/p}\right] \right) =r>0. \end{aligned}
Applying Proposition 1 we conclude that $$\Gamma \left( B\left( 0,r\right) \right) \ge r$$ for all $$0<r<1.$$ From the last estimate it follows in addition that $$\Gamma \left( S\left( 0,1\right) \right) =0.$$

### Corollary 3

For every $$1\le p<\infty ,$$ in the Banach space $$\mathbb {B}=\ell ^{p}$$ there exists a Borel probability measure $$\Gamma$$ with $${\mathrm{supp\,}}\Gamma =B\left( 0,1\right)$$, such that $$\Gamma \left( S\left( 0,1\right) \right) =0$$ and $$\Gamma \left( B\left( 0,r\right) \right) \ge r$$ for all $$0<r<1$$.

A small modification of distributions considered above leads to another interesting situation.

### Example 2

Let $$G_{k},$$$$k\ge 1,$$ be symmetric probability distributions on $$\left[ -1,1\right]$$ satisfying condition (16) for all $$z\in [ r_{1}, 1],$$ and condition (1) for all $$r\in (0,r_{1}),$$ where $$0<r_{1}<1$$ is a fixed number. In other words, we may assume that apart from (16) valid for $$r_{1}\le z\le 1$$, two equal masses
\begin{aligned} G_{k}\left( \left\{ -r_{1}\right\} \right) = \frac{1}{2}\cdot \left\{ 1-\left( 1-r_{1}^{p}\right) ^{2^{k}}\right\} ^{1/p2^{k}} =G_{k}\left( \left\{ r_{1}\right\} \right) \end{aligned}
are assigned to points $$\left\{ -r_{1}\right\} ,$$$$\left\{ r_{1}\right\}$$ by distributions $$G_{k}$$, while $$G_{k}\left( \left[ -z,z\right] \right) =0$$ whenever $$0<z<r_{1}.$$ Then
\begin{aligned} \Gamma \left( B\left( 0,r\right) \right) =0,\quad 0<r<r_{1},\quad \text {and}\quad \Gamma \left( B\left( 0,r\right) \right) \ge r>0,\quad r_{1}\le r<1. \end{aligned}
In consequence, $${\mathrm{supp\,}}\Gamma \subseteq B\left( 0,1\right) \setminus B^{0}\left( 0,r_{1}\right)$$, where $$B^{0}\left( 0,r\right)$$ denotes the open ball $$\{x\in \ell ^p : |x|_{p} < r \}$$.

We leave to the reader further modifications of distributions $$G_{k},$$$$k\ge 1$$, leading to a measure $$\Gamma =P\circ \xi ^{-1}$$ such that $${\mathrm{supp\,}}\Gamma \subseteq B\left( 0,r_{2}\right) \setminus B^{0}\left( 0,r_{1}\right)$$, where $$0<r_{1}<r_{2}<1$$ (cf. remark preceding Sect. 3).

## 4 BRW in Banach Spaces of Martingale Cotype q

The main results given in Sect. 3 for spaces $$\ell ^{p}$$ can be extended to Banach spaces of the same martingale cotype as $$\ell ^{p} .$$ To this end, the first doubt that arises is the question of convergence of the Banach Random Walk (BRW) $$\left\{ Z_{n},n\ge 1\right\}$$ in such Banach spaces. We discuss briefly this problem.

Let $$\mathbb {B}$$ be a Banach space of martingale cotype q for some $$2\le q<\infty ,$$ i.e., there exists a constant C such that for all $$\mathbb {B}$$ -valued martingales $$\left\{ M_{n},n\ge 1\right\}$$ in $$L^{q}\left( \mathbb { B}\right) ,$$
\begin{aligned} \sum \nolimits _{n\ge 1}E\left\| dM_{n}\right\| ^{q}\le C\sup \nolimits _{n\ge 1}E\left\| M_{n}\right\| ^{q}, \end{aligned}
where $$dM_{n}=M_{n}-M_{n-1}$$ for $$n>1$$ and $$dM_{1}=M_{1},$$ see, e.g., [7], Ch. 6, p. 221, and [9], Def. 10.41. By Corollary 4.7, [8], or Corollary 10.7 of [9], there exists a norm $$\left| \cdot \right|$$ equivalent to $$\left\| \cdot \right\|$$ in $$\mathbb {B}$$ such that for a fixed number $$\Delta >0,$$
\begin{aligned} \bigwedge \limits _{x,y\in \mathbb {B}}\quad \left| \frac{x+y}{2} \right| ^{q}+ \Delta \left| \frac{x-y}{2}\right| ^{q}\le \frac{ \left| x\right| ^{q}}{2}+\frac{\left| y\right| ^{q}}{2}, \end{aligned}
which can be rewritten in the form
\begin{aligned} \bigwedge \limits _{x,y\in \mathbb {B}}\quad 1-\left| \frac{x+y}{2} \right| \ge 1-\left( \frac{\left| x\right| ^{q}}{2}+\frac{ \left| y\right| ^{q}}{2}-\Delta \left| \frac{x-y}{2}\right| ^{q}\right) ^{1/q}. \end{aligned}
Therefore
\begin{aligned} \delta \left( \varepsilon \right) =\inf \left\{ 1-\left| \frac{x+y}{2} \right| :\left| x\right| \le 1,\left| y\right| \le 1,\left| x-y\right| \ge \varepsilon \right\} \ge 1-\left( 1-\Delta \left( \frac{\varepsilon }{2}\right) ^{q}\right) ^{1/q}. \end{aligned}
Hence it follows that the space $$\left( \mathbb {B},\left| \cdot \right| \right)$$ is uniformly convex, cf. [7], Th. 6.2, or [9], Th. 10.1 and Prop. 10.31. Since each uniformly convex Banach space is reflexive, cf. Theorem 4.3 of [8], and Theorem 10.3 of [9], taking into account a result of Phillips we conclude that the space $$\left( \mathbb { B},\left| \cdot \right| \right)$$ possesses the RNP, see [4], Ch. III, Sect. 2, Corollary 13, p. 76. Consequently, $$(\mathbb {B}, \Vert \cdot \Vert )$$ also enjoys the RNP.

Assume that $$\left\{ Z_{n},n\ge 1\right\}$$ is a $$\mathbb {B}$$-valued BRW constructed by means of a quasi-orthogonal basis $$\left\{ b_{n},n\ge 1\right\}$$ with respect to $$\Vert \cdot \Vert$$. Applying Lemma 5 of [3] we infer that the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ converges strongly a.s. in $$\left( \mathbb {B} ,\Vert \cdot \Vert \right)$$ and in $$L^{p}\left( \mathbb {B} ,\Vert \cdot \Vert \right)$$ for each fixed $$1\le p<\infty .$$ Now it is evident that all the results given in Sect. 2 are still valid for the Banach space $$\left( \mathbb {B},\left\| \cdot \right\| \right) ,$$ and to generalize the results of Sect. 3 only a small effort is needed.

Having in mind the additional assumption: $$\left( \mathbb {B},\left\| \cdot \right\| \right)$$ is of martingale cotype q$$2\le q<\infty ,$$ we are able to describe convergence of the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ more precisely. Introduce a function $$\left\| \cdot \right\| _{(q)}:\mathbb {B\rightarrow }\left[ 0,\infty \right]$$ given by the formula
\begin{aligned} \left\| x\right\| _{(q)}=\left( \sum \nolimits _{k\ge 1}\left\| x_{k}b_{k}\right\| ^{q}\right) ^{1/q}\quad \text {for}\quad x=\sum \nolimits _{k\ge 1}x_{k}b_{k}\in \mathbb {B}, \end{aligned}
and define $$\mathbb {B}_{q}=\left\{ x\in \mathbb {B}:\left\| x\right\| _{(q)}<\infty \right\} .$$ It can be easily verified that $$\mathbb {B}_{q}$$ is a linear space and $$\left\| \cdot \right\| _{(q)}$$ is a norm in $$\mathbb {B} _{q}$$. (The triangle condition follows from Minkowski’s inequality.) Obviously, $$\{b_n , n\ge 1\}$$ is a quasi-orthogonal, monotone basis in $$(\mathbb {B}_{q} , \left\| \cdot \right\| _{(q)} )$$.
Let $$\widetilde{\mathbb {B}}_q$$ denote the completion of $$\mathbb {B}_q$$ under $$\Vert \cdot \Vert _{(q)}$$. As was already noted, the assumptions imposed in [3] ensure that the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ converges a.s. in $$\left( \mathbb {B},\left\| \cdot \right\| \right)$$ and in $$L^{p}\left( \mathbb {B} ,\left\| \cdot \right\| \right)$$, $$1\le p<\infty$$. Hence it follows that for each $$\varepsilon >0$$ (and every fixed $$1\le p<\infty$$) there can be found $$n_{\varepsilon }$$ such that for all $$m>n\ge n_{\varepsilon },$$ we have $$\left\| \, \left\| Z_{m}-Z_{n}\right\| \, \right\| _{p}<\varepsilon ,$$ where $$\left\| \cdot \right\| _{p}$$ denotes the usual $$L^{p}$$ norm. But for a fixed $$n\ge n_{\varepsilon },$$$$\left\{ Z_{m}-Z_{n},m\ge n\right\}$$ is a martingale, thus in view of Theorem 4.51 [8], or Theorem 10.59 of [9], and the generalized Doob’s inequality, see Corollary 1.13 [8], or Corollary 1.29 [9], we obtain
\begin{aligned} \left\| \left( \sum \limits _{n<k\le m}\left\| dZ_{k}\right\| ^{q}\right) ^{1/q}\right\| _{p}\le & {} C\left\| \sup _{n<k\le m}\left\| Z_{k}-Z_{n}\right\| \right\| _{p}\\\le & {} C\left( p\right) \sup _{m>n}\left\| \left\| Z_{m}-Z_{n}\right\| \right\| _{p}\le C\left( p\right) \varepsilon \end{aligned}
whenever $$1<p<\infty .$$ Consequently, the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ converges also in $$L^{p}\left( \widetilde{\mathbb {B}}_{q},\left\| \cdot \right\| _{(q)}\right)$$ for all $$1<p<\infty .$$ By Theorem 1.14 [8], see also Theorem 2.9 of [9], we conclude in addition that the process $$\left\{ Z_{n},n\ge 1\right\}$$ converges a.s. in $$\left( \widetilde{ \mathbb {B}}_{q},\left\| \cdot \right\| _{(q)}\right)$$. Therefore the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ converges a.s. in the space $$\mathbb {B} \cap \widetilde{\mathbb {B}}_q$$ equipped with norm $$\Vert \cdot \Vert _{\max } = \max \{ \Vert \cdot \Vert , \Vert \cdot \Vert _{(q)}\}$$.
Suppose next that a quasi-orthogonal basis $$\left\{ b_{n},n\ge 1\right\}$$ in a Banach space $$\left( \mathbb {B},\left\| \cdot \right\| \right)$$ is normalized so that $$\left\| b_{n}\right\| =1$$ for all $$n\ge 1.$$ Notice that then
\begin{aligned} \left\| x\right\| _{(q)}=\left( \sum \nolimits _{k\ge 1}\left| x_{k}\right| ^{q}\right) ^{1/q},\quad x=\sum \nolimits _{k\ge 1}x_{k}b_{k}\in \mathbb {B}. \end{aligned}
In such a case the spaces $$\left( \widetilde{\mathbb {B}}_{q},\left\| \cdot \right\| _{(q)}\right)$$ and $$\ell ^{q}$$ are isometrically isomorphic, and thus we may identify $$\widetilde{\mathbb {B}}_{q}$$ with $$\ell ^{q}.$$ Therefore the main results of Sect. 3, in particular Proposition 1 and Corollary 3, remain valid provided the space $$\ell ^{q}$$ is replaced by $$\left( \widetilde{\mathbb {B}} _{q},\left\| \cdot \right\| _{(q)}\right)$$. In this way we obtain the following result.

### Theorem 3

Let $$\left( \mathbb {B},\left\| \cdot \right\| \right)$$ be a Banach space of martingale cotype q for some $$2\le q<\infty ,$$ with a quasi-orthogonal Schauder basis $$\left\{ b_{n},n\ge 1\right\}$$ normalized so that $$\left\| b_{n}\right\| =1$$, $$n\ge 1.$$ Moreover, let $$\left\{ G_{n},n\ge 1\right\}$$ be a sequence of symmetric probability distributions on the interval $$\left[ -1,1\right]$$ satisfying conditions of Proposition 1 with p replaced by q. Then for a fixed $$0<r<1$$, we have
\begin{aligned} \Gamma \left( B_{q}\left( 0,r\right) \right) \ge c_{r}>0, \end{aligned}
where $$B_{q}\left( 0,r\right) =\left\{ x\in \mathbb {B}: \left\| x\right\| _{(q)}\le r\right\}$$, $$0<r<\infty ,$$$$\Gamma =P\circ \xi ^{-1},$$ and $$\xi$$ is the a.s. limit of the BRW $$\left\{ Z_{n},n\ge 1\right\}$$ in $$\mathbb {B}\cap \widetilde{\mathbb {B}}_q$$ generated by $$\left\{ G_{n},n\ge 1\right\} .$$ Hence it follows that the whole mass of the measure $$\Gamma$$ is not concentrated on the set $$S_{q}\left( 0,1\right) =\left\{ x\in \mathbb {B} :\left\| x\right\| _{(q)}=1\right\} .$$

As a consequence of this approach and Corollary 3 we get

### Corollary 4

For every Banach space $$\left( \mathbb {B} ,\left\| \cdot \right\| \right)$$ of martingale cotype $$2\le q<\infty ,$$ with a quasi-orthogonal normalized Schauder basis $$\left\{ b_{n},n\ge 1\right\} ,$$ there exists a Borel probability measure $$\Gamma$$ with $${\mathrm{supp\,}}\Gamma = B_{q}\left( 0,1\right) ,$$ such that $$\Gamma \left( S_{q}\left( 0,1\right) \right) =0$$ and $$\Gamma \left( B_{q}\left( 0,r\right) \right) \ge r$$ for all $$0<r<1.$$

## Notes

### Acknowledgements

The authors are grateful to the referee for helpful remarks which led to substantial improvement of the paper.

## References

1. 1.
Banach, S.: The Lebesgue integral in abstract spaces. In: Saks, S. (ed.) Theory of the Integral. Dover Publication, New York (1964)Google Scholar
2. 2.
Banek, T.: Banach random walk in the unit ball $$S\subset \ell ^{2}$$ and chaotic decomposition of $$\ell ^{2}\left( S,\mathbb{P}\right)$$. J. Theor. Probab. 29, 1728–1735 (2016)
3. 3.
Banek, T., Zapała, A.M.: Random walk in balls and an extension of the Banach integral in abstract spaces. J. Theor. Probab. (submitted, 2015)Google Scholar
4. 4.
Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence, RI (1977)
5. 5.
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1971)
6. 6.
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, II. Springer, Berlin (1977, 1979)Google Scholar
7. 7.
Pisier, G.: Probabilistic methods in the geometry of Banach spaces. Lect. Notes Math. 1206, 167–241 (2006)
8. 8.
Pisier, G.: Martingales in Banach Spaces (in connection with Type and Cotype). Course IHP (Feb 2-8, 2011) February 9 (2011). https://webusers.imj-prg.fr/~gilles.pisier/ihp-pisier.pdf
9. 9.
Pisier, G.: Martingales in Banach Spaces. Cambridge University Press, Cambridge (2016)
10. 10.
Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A.: Probability Distributions on Banach Spaces. D. Reidel, Dordrecht (1987)