Convergence of multidimensional random sequences with independent random indices (Q1917609)

The weak convergence of \(\mathbb R^m\)-valued random sequences with independent random indices under operator normalization is investigated. Assume that \(\{Y_{n,k}\}_{k\geq 1}\), \(n=1,2,\dots\), are sequences of random elements taking values in an \(m\)-dimensional Euclidean space \(\mathbb R^m\). Let \(\Theta(\mathbb R^m)\) be the set of linear nonsingular operators acting from \(\mathbb R^m\) to \(\mathbb R^m\), \(\{B_{n,k}\}_{k\geq 1}\) a sequence of operators from \(\Theta(\mathbb R^m)\), and \(\{a_{n,k}\}_{k\geq 1}\), \(a_{n,k}\in\mathbb R^m\), \(n=1,2,\dots\), a sequence of nonrandom elements. Define for fixed random elements \(Z\) and \(Y\) \[ W(Z\mid Y)=\{W(g),\;g\in\mathbb R^m: Z=W(Y),\;W(g)\text{ and }Y\text{ are independent}\}, \] and introduce the functional \[ \Lambda_H(X_1(\cdot),X_2(\cdot))= \int_{\mathbb R^m}\Pi(X_1(g),X_2(g)) H(dg), \] where \(\Pi\) is the Lévy-Prokhorov metric, \(X_1(g)\) and \(X_2(g)\) measurable \(\mathbb R^m\)-valued random fields and \(H\) some distribution defined on \({\mathcal B}(\mathbb R^m)\). The main result is as follows. Let the families of r.e.'s \(\{Y_{n,k}\}_{k\geq 1,n\geq 1}\) and r.v.'s \(\{|D^{-1}_n B_{n,N_n}|\}_{n\geq 1}\), \(D_n\in\Theta(\mathbb R^m)\), be tight and \(Y_{n,k}\Rightarrow Y_n\) as \(k\to\infty\) for some \(Y_n\) and any \(n\geq 1\) so that the compatibility condition (see the paper) holds. Assume that there exists a r.e. \(Y\) with distribution \(H\) such that \(Y_n\Rightarrow Y\) as \(n\to\infty\). Then \(Z_n\Rightarrow\) (some) \(Z\) \((n\to\infty)\) for some sequence \(\{c_n\}_{n\geq 1}\), \(c_n\in\mathbb{R}^m\), if and only if there exists a \(\Lambda_H\)-compact sequence of measurable random fields \(\{W_n'(\cdot)\}_{n\geq 1}\), \(W_n'(\cdot)\in {\mathcal W}(Z\mid Y)\) \((n\geq 1)\) such that \(\lim_{n\to\infty}(W_n'(\cdot),W_n(\cdot))=0\).