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Zapiski Nauchnykh Seminarov POMI, 2023, Volume 525, Pages 109–121 (Mi znsl7371)  

On complete convergence of moments of i.i.d.r.v. with finite variances

L. V. Rozovsky

Saint-Petersburg State Chemical-Pharmaceutical University
References:
Abstract: Let $\{X_n\}, n\ge 1,$ be a sequence of independent random variables with common distribution functions, zero means and unit variances, $\bar{S}_n =( X_1 +\cdots + X_n)/\sqrt n$. The main goal of this note is a study of the behavior of sums
$$ \Sigma_r(\varepsilon) = \sum\limits_{n\ge 1} n^s \mathbf{E} \bar S^r_n I[\bar S_n\ge \varepsilon n^\delta], $$
as $\varepsilon\to +0$ under optimal (that is, necessary) moment assumptions, where $\delta, s, r$ are some constants, such that $\delta> 0$ and $s+1$ and $r$ are non-negative. In particular, it is shown that if $s>-1/2$ and $(2-r) \delta = s+1$, then
$$ \varepsilon^{2-r} \Sigma_r(\varepsilon) = \dfrac{1}{2\delta (2-r)} + O \big(\lambda(\rho)\big),\ \rho=\varepsilon^{-1/2\delta}, \lambda(\rho)=\mathbf{E} X_1^2 \Big(1 \land \dfrac{| X_1|}{\rho}\Big). $$
A similar estimate with a more complicated formulation holds also in the case $-1<s\le -1/2$. Thus, for $\delta=1/2$ we generalize the pioneering result of Heyde (Appl. Probab., 1975) and most its refinements (e.g. due to He and Xie (Acta Math. Appl. Sin., 2013)), as well as the corresponding statements of Liu and Lin (Statist. Probab. Lett. 2006) and Kong and Dai (Stoch. Dynamics, 2017).
Key words and phrases: convergence rate, exact asymptotics, complete convergence of moments.
Received: 03.07.2023
Document Type: Article
Language: Russian
Citation: L. V. Rozovsky, “On complete convergence of moments of i.i.d.r.v. with finite variances”, Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, POMI, St. Petersburg, 2023, 109–121
Citation in format AMSBIB
\Bibitem{Roz23}
\by L.~V.~Rozovsky
\paper On complete convergence of moments of i.i.d.r.v. with finite variances
\inbook Probability and statistics. Part~34
\serial Zap. Nauchn. Sem. POMI
\yr 2023
\vol 525
\pages 109--121
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl7371}
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