Functional central limit theorem for random multilinear forms
Let \(X_ 1, X_ 2,\dots\) be a sequence of independent random variables (r.v.'s) with \[ EX_ j=0, \quad EX^ 2_ j=1, \quad j=1,2, \dots \tag{1} \] Let \(Q_{n,k} = 0\) for \(n<k\), and \(Q_{n,k} = \sum_{1 \leq i_ 1 \neq \cdots \neq i_ k \leq n} a_{i_ 1 \dots i_ k} X_{i_ 1} \dots X_{i_ k}\) for \(n \geq k\). Denote \(B^ 2_ n=DQ_{n,k} = \sum_{1 \leq i_ 1 \neq \cdots \neq i_ k \leq n} a^ 2_{i_ 1 \dots i_ k}\). Define a random piecewise linear function on \([0,1]\) by \[ \xi_ n (t)={1 \over B_ n} \left( Q_{r,k} + {(Q_{r+1,k} - Q_{r,k}) (tB^ 2_ n-B^ 2_ r) \over B^ 2_{r+1}-B^ 2_ r} \right),\qquad t \in \left[ {B^ 2_ r \over B^ 2_ n}, {B^ 2_{r+1} \over B^ 2_ n} \right],\;r=0,1, \dots,n-1. \] Theorem 1. Let \(X_ 1, X_ 2, \dots\) be a sequence of r.v.'s satisfying (1). If certain conditions are satisfied, then for all \(x \in \mathbb{R}\) \[ \lim_{n \to \infty} P\{Q_{n,k}/B_ n<x\} = \Phi (x), \tag{2} \] where \(\Phi(x)\) is the standard normal distribution function. Furthermore, all finite- dimensional distributions of \(\xi_ n (t)\) converge weakly to those of \(W(t)\), where \(W(t)\) denotes a standard Brownian motion on \([0,1]\).
scientific article; zbMATH DE number 578643