Limit Distribution of the Banach Random Walk

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 $\Gamma$.

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

$$\begin{aligned}&\left\| \frac{\pi _{n-1}\left( x\right) }{r}+\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}-\frac{\pi _{n-1}\left( x\right) }{r} -\alpha _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \\&\quad =2\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) \cdot \left\| b_{n}\right\| >2\cdot \frac{1}{\left\| b_{n}\right\| } \cdot \left\| b_{n}\right\| =2, \end{aligned}$$

which leads to a contradiction with the conditions

$$\begin{aligned} \left\| \frac{\pi _{n-1}\left( x\right) }{r}+\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \le 1, \quad \left\| \frac{\pi _{n-1}\left( x\right) }{r} +\alpha _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \le 1 . \end{aligned}$$

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 $0<r<1.$