On the necessity of Lindeberg’s condition in the weak invariance principle for functions of independent variables. (English. Russian original) Zbl 0627.60040

Vlasenko, S. N.

Theory Probab. Math. Stat. 34, 47-50 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 44-47 (1986).

Let \(\{\xi_{in}\), \(1\leq i\leq n\), \(n\geq 1\}\) be a sequence of sequences of random variables independent in each row, and let \(\{\eta_{in}\), \(1\leq i\leq n,n\geq 1\}\) be another such sequence which is also independent of the first sequence. For a sequence \(f_ n: {\mathbb{R}}^ n\to {\mathbb{R}}\) of Borel functions, conditions for \(f_ n(\xi_{1n},...,\xi_{nn})\) and \(f_ n(\eta_{1n},...,\eta_{nn})\) to have asymptotically the same distribution are found. It is shown that for a class of these \(f_ n's\), a Lindeberg type condition is necessary and sufficient for such an equivalence. A corollary of the result for nonnegative definite quadratic forms is formulated.

Reviewer: M.M.Rao

MSC:

60F17

Functional limit theorems; invariance principles

Keywords:

weak invariance principle; Lindeberg type condition; nonnegative definite quadratic forms