### Abstract

For a sequence of i.i.d. mean 0 random variables {X, X_{n} ; n ≥ 1} with weighted partial sums S_{n} (X, w ({dot operator})) = ∑_{k = 1}^{n} w (frac(k, n)) X_{k}, n ≥ 1 where w (t), 0 ≤ t ≤ 1 is a Lipschitz function of order 1 with {norm of matrix} w ({dot operator}) {norm of matrix}_{2} = sqrt(∫_{0}^{1} w^{2} (t) d t) > 0, necessary and sufficient conditions are provided for X to enjoy iterated logarithm type behavior of the form 0 < lim sup_{n → ∞} | S_{n} (X, w ({dot operator})) | / sqrt(n h (n)) < ∞ almost surely where h ({dot operator}) is a positive, nondecreasing function which is slowly varying at infinity. Some corollaries are presented for particular choices of h ({dot operator}).

Original language | English (US) |
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Pages (from-to) | 643-651 |

Number of pages | 9 |

Journal | Statistics and Probability Letters |

Volume | 79 |

Issue number | 5 |

DOIs | |

State | Published - Mar 1 2009 |

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## Cite this

*Statistics and Probability Letters*,

*79*(5), 643-651. https://doi.org/10.1016/j.spl.2008.10.010