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Journal of Theoretical Probability

, Volume 32, Issue 1, pp 47–63 | Cite as

Limit Distribution of the Banach Random Walk

  • Tadeusz Banek
  • Patrycja Jędrzejewska
  • August M. ZapałaEmail author
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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
$$\begin{aligned}&\left\| \frac{\pi _{n-1}\left( x\right) }{r}+\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}-\frac{\pi _{n-1}\left( x\right) }{r} -\alpha _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) b_{n}\right\| \\&\quad =2\beta _{n}\left( \frac{\pi _{n-1}\left( x\right) }{r}\right) \cdot \left\| b_{n}\right\| >2\cdot \frac{1}{\left\| b_{n}\right\| } \cdot \left\| b_{n}\right\| =2, \end{aligned}$$
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 \(0<r<1. \)

Notes

Acknowledgements

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

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Tadeusz Banek
    • 1
  • Patrycja Jędrzejewska
    • 2
  • August M. Zapała
    • 2
    Email author
  1. 1.Pope John Paul II State School of Higher Education in Biała PodlaskaBiała PodlaskaPoland
  2. 2.Faculty of Mathematics, Informatics and Landscape ArchitectureThe John Paul II Catholic University of LublinLublinPoland

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