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This article is cited in 31 scientific papers (total in 32 papers)
On almost sure limit theorems
I. A. Ibragimova, M. A. Lifshitsb a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Saint-Petersburg State University
Abstract:
By a sequence of random vectors $\{\zeta_k\}$, we can construct empirical distributions of the type $ Q_n = (\log n)^{-1} \sum_{k=1}^n \delta_{\zeta_k}/k$. Statements on the convergence of these or similar distributions with probability 1 to a limit distribution are called almost sure theorems. We propose several methods which permit us to easily deduce the almost sure limit theorems from the classical limit theorems, prove the invariance principle of the type “almost sure” and investigate the convergence of generalized moments. Unlike the majority of the preceding papers, where only convergence to the normal law is considered, our results may be applied in the case of limit distributions of the general type.
Keywords:
limit theorems, convergence almost sure, sums of independent variables, weak dependence, invariance principle.
Received: 12.02.1998
Citation:
I. A. Ibragimov, M. A. Lifshits, “On almost sure limit theorems”, Teor. Veroyatnost. i Primenen., 44:2 (1999), 328–350; Theory Probab. Appl., 44:2 (2000), 254–272
Linking options:
https://www.mathnet.ru/eng/tvp767https://doi.org/10.4213/tvp767 https://www.mathnet.ru/eng/tvp/v44/i2/p328
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