## Abstract

Let X
_{1}, X
_{2}, … be independent random variables with EX
_{k}= 0 and σk2:=EXk2<∞(k≥ 1) . Set S
_{k}= X
_{1}+ ⋯ + X
_{k} and assume that sk2:=ESk2→∞ . We prove that under the Kolmogorov condition |Xn|≤Ln,Ln=o(sn/(loglogsn)1/2) we have 1logsn2∑k=1nσk+12sk2f(Sksk)→12π∫Rf(x)e-x2/2dxa.s. for any almost everywhere continuous function f: R→ R satisfying |f(x)|≤eγx2 , γ< 1 / 2 . We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process {Sn,n≥1} by a Wiener process.

Original language | English |
---|---|

Number of pages | 19 |

Journal | Journal of Theoretical Probability |

Early online date | 6 May 2023 |

DOIs | |

Publication status | E-pub ahead of print - 6 May 2023 |

## Keywords

- Almost sure central limit theorem
- Sums of independent random variables
- Weighted averages

## ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty