Invariance principle for independent observations of random processes with values in Banach spaces
Let \(Y_1, Y_2,\dots\) be i.i.d. random processes defined on a probability space, whose expectation operator will be denoted by \(E\), with values in a Banach space \(B\) and parameter in \([0,1]\) (\(Y_i\) is assumed product measurable). Moreover, let \(\xi_1, \xi_2,\dots\) be i.i.d. random variables with values in \([0,1]\) defined on a second probability space with expectation operator \(E'\). One of the results of the paper states that if \(B\) is of type 2, \(Y_1\) is centered and \(E'E\|Y_1(\xi_1)\|^2<\infty\), then the random polygonal function in \(C([0,1],B)\) (\(B\)-valued continuous functions on \([0,1]\)) with vertices at \((\frac kn, n^{-1/2}\sum_{i\leq k}Y_i(\xi_i(\omega')))\) converges in distribution, for almost all \(\omega'\) in the second probability space, to the Gaussian \(B\)-valued random process \(W\) on \([0,1]\) with independent increments such that \(W(t)\) has the law of \(t^{1/2}G\), \(t\in[0,1]\), \(G\) having the covariance \(E'Ex(Y_1(\xi_1))y(Y_1(\xi_1))\), \(x\), \(y\) in the dual of \(B\). This continues previous work of the author [for example, see Theory Probab. Appl. 41, No. 4, 756-761 (1996); translation from Teor. Veroyatn. Primen. 41, No. 4, 914-919 (1996; Zbl 0895.60003)].
scientific article; zbMATH DE number 1541461