Abstract
Considering a sequence of i.i.d. positive random variables, for products of sums of partial sums we establish an almost sure central limit theorem, which holds for some class of unbounded measurable functions.
Highlights
Introduction and main resultsLet {Xn; n ≥ } be a sequence of random variables and define Sn = n i= XiSome results as regards the limit theorem of products n j=Sj were obtained in recent years.Rempala and Wesolowski [ ] obtained the following asymptotics for products of sums for a sequence of i.i.d. random variables.Theorem A Let {Xn; n ≥ } be a sequence of i.i.d. positive square integrable random variables with EX = μ, the coefficient of variation γ = σ /μ, where σ = Var(X )
Tan and Peng [ ] proved the result of Theorem B still holds for some class of unbounded measurable functions and obtained the following result
In order to prove Theorem . , we introduce the following lemmas
Summary
Wesolowski [ ] obtained the following asymptotics for products of sums for a sequence of i.i.d. random variables. And in the sequel, N is a standard normal random variable and →d denotes the convergence in distribution. Tan and Peng [ ] proved the result of Theorem B still holds for some class of unbounded measurable functions and obtained the following result.
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