Banach Random Walk in the Unit Ball \(S\subset l^{2}\) and Chaotic Decomposition of \(l^{2}\left( S,{{\mathbb {P}}}\right) \)

Journal of Theoretical Probability, Jun 2015

A Banach random walk in the unit ball S in \(l^{2}\) is defined, and we show that the integral introduced by Banach (Theory of the integral. Warszawa-Lwów, 1937) can be expressed as the expectation with respect to the measure \({{\mathbb {P}}}\) induced by this walk. A decomposition \(l^{2}\left( S,{{\mathbb {P}}}\right) =\bigoplus _{i=0}^{\infty } {{\mathfrak {B}}}_{i}\) in terms of what we call Banach chaoses is given.

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Banach Random Walk in the Unit Ball \(S\subset l^{2}\) and Chaotic Decomposition of \(l^{2}\left( S,{{\mathbb {P}}}\right) \)

Mathematics Subject Classification Banach Random Walk in the Unit Ball S ⊂ l 2 and Chaotic Decomposition of l 2 ( S, P) Tadeusz Banek 0 0 Faculty of Management, Lublin University of Technology , Nadbystrzycka 38, 20-618 Lublin , Poland A Banach random walk in the unit ball S in l2 is defined, and we show that the integral introduced by Banach (Theory of the integral. Warszawa-Lwów, 1937) can be expressed as the expectation with respect to the measure P induced by this walk. A decomposition l2 (S, P) = i∞=0 Bi in terms of what we call Banach chaoses is given. 1 Introduction We propose an averaging procedure based on Banach’s concept of Lebesgue integral in abstract spaces [1]. To be specific, we are going to use a particular variant of Banach’s theory, connected with integration in l2. We denote by k=1 Sn = x ∈ Rn : xk2 ≤ 1 and S = x ∈ RN : k=1 xk2 ≤ 1 the unit balls in ln2 and l2, respectively. According to Banach’s result, the most general nonnegative linear functional defined on S and satisfying certain conditions listed in [1] (which we do not need to repeat here) has the form B Tadeusz Banek g (x1) g x2/ 1 − x12 . . . g xn/ 1 − x1 − · · · − xn−1 2 2 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 and g : [−1, 1] → [0, ∞), −11 g (t ) dt = 1, Φ : RN → R is a bounded Borel measurable function, and χA is the indicator of A. Although Banach’s considerations and constructions are purely deterministic and based on ideas coming from functional analysis, his expression of ρn can be easily reinterpreted in probabilistic terms, giving a probabilistic interpretation for his extension of Lebesgue integral. The first step in this direction is to find a stochastic sequence having probability density function ρn. Such a sequence will be called a Banach random walk (BRW), or a standard Banach random walk (SBRW) if g ≡ 1. The expression of Fn (Φ) in terms of BRW is immediate. In Sects. 3 and 4, an orthogonal expansion of square integrable functionals of the BRW [elements of l2 (Sn)] in terms of Legendre polynomials is obtained, and a chaotic decomposition of l2 (S) is presented. These are the main results of this paper. 2 Banach Random Walk on Sn (x1, 0) is in S2. Choose x2 randomly in 2 2 − 1 − x1 , 1 − x1 with density − 1 − x12 − x22, 1 − x12 − x22 with density g x3/ 1−x12−x22 / 1−x12 − x22, etc. The sequence x1, . . . , xn is random, and the probability density function corresponding to this sample is ρn (x1, . . . , xn), as in the Banach integral. To check that it is a density, it is enough to show that In = ⎢ Sn−1 ⎣ − 1−x12−···−xn2−1 = In−1 = I1 = g (x1) dx1 = 1. −1 3 Legendre Polynomials Legendre polynomials in one variable are defined by the formulae L p (t ) = The polynomials are orthogonal: 2−1 −1 L p (t ) Lq (t ) dt = and L p (·) : p = 0, 1, . . . is a complete set in L2 [−1, 1]. To extend these to the multivariate case, we introduce a mapping Θ : Sn → [−1, 1]n by 2 y2 = Θ2 (x) = x2/ 1 − x1 , x2 = Θ2−1 (y) = y2 1 − y1 , 2 yn = Θn (x) = xn/ 1 − x1 − · · · − xn2−1, xn = Θn−1 (y) = yn 2 = 2−n [−1,1]n 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 Ψn (y) dy (where Ψn = Φn ◦ Θ−1). For a multi-index p = ( p1, p2, . . .), define Ln, p (y) = i=1 2−n [−1,1]n Ln, p (y) Ln,q (y) dy = if p = q, i=1 ln, p (y) = p∈Nn0 [−1,1]n is an orthonormal basis for L2 [−1, 1]n , 2−n dx , and any element Ψn of this space has a unique orthogonal expansion 4 Orthogonal Decomposition of l 2 (S, P) The orthogonal decomposition of spaces of square integrable random variables dates back to Wiener [2] and was continued by Ito [3] for the continuous-time counterpart of SBRW, which is the standard Wiener process. These results were applied to diffusion processes in [4,5] and were recently extended to Lévy processes (see [6,7] for instance). This line of research has several motivations, beginning with usefulness of orthogonal representations for approximation and ending with applications in Malliavin calculus (see [8,9] for instance) and stochastic analysis in general (see [10]). Our situation is different since BRW is neither Gaussian nor Markov. Nevertheless, it appeared naturally in Banach’s extension of Lebesgue integral to abstract spaces. It is worth mentioning that Banach’s method uses functional analytic tools and “is not based on the notion of measure” (according to [1]). 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 < ∞, and we say that Φ ∈ l2 (S) if Φ : S → R, and Φ l22(S) = nl→im∞ Sn 2n 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 < ∞, Φn (x ) = Φ (x1, . . . , xn , 0, . . .) = (Φ ◦ πn) (x ) , → Rn is the projection onto the first n coordinates. Definition 2 Let Ω = [−1, 1]N, F = n∞=1 B ([−1, 1]), where B ([−1, 1]) is the Borel sigma field on [−1, 1], and P = n∞=1 21 λ[−1,1], where λ[−1,1] is the onedimensional Lebesgue measure restricted to [−1, 1]. On (Ω, F , P) define Y = (Y1, Y2, . . .), where Yi (ω) = ωi , ω = (ω1, ω2, . . .) ∈ Ω, i.e., Yi is a sequence of i.i.d. random variables uniformly distributed on [−1, 1], and X = (X1, X2, . . .), where = E ! f ◦ Θ−1 Y n " = 2−n [−1,1]n 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 Definition 3 We say that a random variable F : Ω → R belongs to the space l2 (Sn, Pn) [(respectively, l2 (S, P)] if it is of the form F = Φn(X n) (resp. F = Φ(X)) F l22(Sn,Pn) = EPn Φn X n 2 < ∞, (respectively, F l22(S,P) = EP [Φ (X)]2 < ∞). Theorem 1 If Φ ∈ l2 (S), then Φ (X) = nl→im∞ Φn X n , in the norm l2 (S, P) l2 (S, P) = i=0 Bi = p∈Nn0 i=2 where spanl pi (·) stands for the closure in l2 (S, P). Proof Since l2 (S, P) is a Hilbert space, to prove the first equality, it is enough to show that Φn (X n), n = 1, 2, . . . , is a Cauchy sequence. Indeed, if n ≤ m, we have Φn (X n) = Φm (X n, 0, . . . , 0), hence p∈N0m\N0n×{0}m−n i=2 1 − (X1)2 − · · · − (Xi−1)2 where ψp = 0 for all multi-indices p = ( p1, . . . , pn, 0, . . .); hence, by orthonormality of l pi , we get En Φn X n − Φm X m 2 = p∈N0m\N0n×{0}m−n 1 − x12 . . . 1 − x12 − · · · − x 2j−1 [−1,1] l p j y j dy j = 0, 1 − (X1)2 − · · · − X j−1 1 − (X1)2 − · · · − (Xi−1)2 0 = pi ,q j ∈N0 = E Bi B j 1 − (X1)2 − · · · − X j−1 for i = j . The crucial argument used in the proof above will be repeated below to show stochastic independence of the renormalized walk. Proposition 2 If X = (X1, X2, . . .) is a SBRW on some probability space (Ξ, , Q), then the random variables Yn = 1 − (X1)2 − · · · − (Xn−1)2 , n = 1, 2, . . . , are stochastically independent with uniform distribution on [−1, 1]. Consequently, all the Banach chaoses Bi , i = 0, 1, . . . , are stochastically independent. Proof Indeed, for every Borel bounded measurable f : R → R and g : R → R, we have 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 (n > m) [−1,1]n f (yn) g (ym) dy 2−1 f (yn) dyn 2−1g (ym) dym 1 − x12 . . . 1 − x1 − · · · − xn−1 2 2 1 − x12 . . . 1 − x1 − · · · − xm−1 2 2 = EQ [ f (Yn)] EQ [g (Ym)] . = 2−n [−1,1] Remark 1 For a purely deterministic mathematical object, namely a linear, nonnegative functional on l2 (S) expressed in the form of Banach’s extension of Lebesgue integral, we found a deeply hidden random object, namely a SBRW, which is closely connected with it and can be used for its equivalent representation. 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Tadeusz Banek. Banach Random Walk in the Unit Ball \(S\subset l^{2}\) and Chaotic Decomposition of \(l^{2}\left( S,{{\mathbb {P}}}\right) \), Journal of Theoretical Probability, 2016, 1728-1735, DOI: 10.1007/s10959-015-0620-1