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Central limit theorems revisited

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<mark>Journal publication date</mark>15/04/2000
<mark>Journal</mark>Statistics and Probability Letters
Issue number3
Volume47
Number of pages11
Pages (from-to)265-275
Publication StatusPublished
<mark>Original language</mark>English

Abstract

A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale difference arrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a (possibly degenerate) constant. This result is then extended, first to multi-dimensions and next to Hilbert-valued elements, under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindeberg condition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued random elements is indicated.