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On the complete convergence of moments in exact asymptotics under normal approximation
L. V. Rozovskii Saint-Petersburg State Chemical-Pharmaceutical University
Abstract:
For the sums of the form $\overline I_s(\varepsilon) = \sum_{n\geqslant 1}
n^{s-r/2}\mathbf{E}|S_n|^r\,\mathbf I[|S_n|\geqslant \varepsilon\,n^\gamma]$,
where $S_n = X_1 +\dots + X_n$, $X_n$, $n\geqslant 1$, is a sequence of
independent and identically distributed random variables (r.v.'s) $s+1
\geqslant 0$, $r\geqslant 0$, $\gamma>1/2$, and $\varepsilon>0$, new results
on their behavior are provided. As an example, we obtain the following
generalization of Heyde's result [J. Appl. Probab., 12 (1975),
pp. 173–175]: for any $r\geqslant 0$, $\lim_{\varepsilon\searrow
0}\varepsilon^{2}\sum_{n\geqslant 1} n^{-r/2} \mathbf{E}|S_n|^r\,\mathbf
I[|S_n|\geqslant \varepsilon\, n] =\mathbf{E} |\xi|^{r+2}$ if and only if
$\mathbf{E} X=0$ and $\mathbf{E} X^2=1$, and also
$\mathbf{E}|X|^{2+r/2}<\infty$ if $r < 4$, $\mathbf{E}|X|^r<\infty$ if $r>4$,
and $\mathbf{E} X^4 \ln{(1+|X|)}<\infty$ if $r=4$.
Here, $\xi$ is a standard Gaussian r.v.
Keywords:
convergence rate, exact asymptotics, complete convergence of moments.
Received: 02.02.2023 Revised: 23.04.2023 Accepted: 16.02.2023
Citation:
L. V. Rozovskii, “On the complete convergence of moments in exact asymptotics under normal approximation”, Teor. Veroyatnost. i Primenen., 68:4 (2023), 769–778; Theory Probab. Appl., 68:4 (2024), 622–629
Linking options:
https://www.mathnet.ru/eng/tvp5634https://doi.org/10.4213/tvp5634 https://www.mathnet.ru/eng/tvp/v68/i4/p769
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Abstract page: | 75 | Full-text PDF : | 1 | References: | 21 | First page: | 7 |
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