### Abstract

This note is devoted to a generalization of the Strassen converse. Let gn:R ∞→[0,∞], n≥1 be a sequence of measurable functions such that, for every n≥1, gn(x+y2)≤C(gn(x)+gn(y)) and gn(x-y2)≤C(gn(x)+gn(y)) for all x,y R∞, where 0<C<∞ is a constant which is independent of n. Let {X,Xn;n≥1} be a sequence of i.i.d. random variables. Assume that there exist r≥1 and a function φ:[0,∞)→[0,∞) with limt→∞φ(t)=∞, depending only on the sequence {gn;n≥1} such that lim supn→∞gn(X1,X2,∞)=φ(E|X|r) a.s. whenever E|X|r<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X1,X2,∞)<∞ a.s. implies E|X|r<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,∞)=∞ for all c≠0). Some applications are provided to illustrate this result.

Original language | English (US) |
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Pages (from-to) | 729-735 |

Number of pages | 7 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 374 |

Issue number | 2 |

DOIs | |

State | Published - Feb 15 2011 |

### Keywords

- Convergence of generalized moments
- I.i.d. random variables
- The Strassen converse
- The law of the iterated logarithm

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*374*(2), 729-735. https://doi.org/10.1016/j.jmaa.2010.09.020