On the convergence of random polynomials and multilinear forms (Q719487)

Given a positive integer \(k\), a multi-indexed sequence of complex number \((a_{j_1,\dots,j_k})\) is said to be symmetric if \(a_{j_1,\dots,j_k}=a_{l_1,\dots,l_k}\) for all permutations \((l_1,\dots,l_k)\) of \((j_1,\dots,j_k)\), while a sequence of complex valued random variables \((X_j)_j\) is said to satisfy condition (*) if \(\inf_{j\in{\mathbb N}}\text{E}(|X_j|)>0\) and \(\sup_{j\in{\mathbb N}}\text{E} (|X_j|^{2k})<\infty\). The authors prove that, if \((a_{j_1,\dots,j_k})\) is a symmetric sequence of complex numbers and \((X_j)_j\) is a sequence of independent and rotation invariant random variables which satisfies condition (*), then \(\sum_{j_1,\dots,j_k}^na_{j_1,\dots,j_k}X_{j_1}\dots X_{j_k}\) converges almost surely if and only if \(\sum_{j_1,\dots,j_k}^n|a_{j_1,\dots,j_k}|^2<\infty\). A complex valued random variable is said to be Steinhaus if it can be written in the form \(e^{i\phi}\) where \(\phi\) is a uniform random variable on the interval \([0,2\pi]\). It is shown that, if \((a_{j_1,\dots,j_k})\) is a symmetric sequence of complex numbers and \((X_j)_j\) is a sequence of Steinhaus random variables, then \(\sum_{j_1,\dots,j_k}^na_{j_1,\dots,j_k}X_{j_1}\dots X_{j_k}\) converges in \(L_p\) for every \(p\) with \(1\leq p<\infty\) if and only if \(\sum_{j_1,\dots,j_k}^n| a_{j_1,\dots,j_k}|^2<\infty\). Given an injective Hilbert-Schmidt operator \(T\) from \(\ell_2\) into \(\ell_2\), the completion of the space of finite sequences with the norm \(||| x|||=\|Tx\|_{\ell_2}\) is denoted by \(X_T\) and called a standard full subspace. The authors prove that, if \((a_{j_1,\dots,j_k})\) is a symmetric sequence of complex numbers and \((X_j)_j\) is a sequence of independent and rotation invariant random variables, then the series \(\sum_{j_1,\dots,j_k}^na_{j_1,\dots,j_k} X_{j_1}\dots X_{j_k}\) converges in a standard full subspace if and only if the polynomial \(P(x)=\sum_{j_1,\dots,j_k}^na_{j_1,\dots,j_k}x_{j_1}\dots x_{j_k}\) defines a 2-dominated \(k\)-homogeneous polynomial on \(\ell_2\). Each of the above results is shown to be equivalent to their \(k\)-linear formalisation. Among the techniques the authors use to obtain their results are Gaussian random variables and the theory of summing operators on Banach spaces.