### Abstract

Let X_{1}, X_{2}, . . . be i.i.d. r.v.'s with EX_{1} = 0, EX^{2}_{1} = 1 and put S_{n} = X_{1} + ··· + X_{n}. We investigate the a.s. limiting behavior of (equation presented) for general norming sequences (a_{k}). The pointwise central limit theorem shows that L_{N} converges a.s. to the normal distribution if a_{k} - √k; in our paper we prove the surprising result that for suitably chosen (a_{k}) the expression L_{N} can converge also to non-Gaussian limits, in particular, any symmetric stable distribution is a possible limit of L_{N}. We shall determine the class of limit distributions of L_{N} and extend the result to the case when X_{n} belong to the domain of attraction of a stable law.

Original language | English |
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Pages (from-to) | 361-368 |

Number of pages | 8 |

Journal | Statistics and Probability Letters |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - Sep 16 1996 |

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### Keywords

- Mixtures
- Pointwise central limit theorem
- Stable distributions

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

**On the pointwise central limit theorem and mixtures of stable distributions.** / Berkes, I.; Csáki, E.

Research output: Contribution to journal › Article

*Statistics and Probability Letters*, vol. 29, no. 4, pp. 361-368. https://doi.org/10.1016/0167-7152(95)00192-1

}

TY - JOUR

T1 - On the pointwise central limit theorem and mixtures of stable distributions

AU - Berkes, I.

AU - Csáki, E.

PY - 1996/9/16

Y1 - 1996/9/16

N2 - Let X1, X2, . . . be i.i.d. r.v.'s with EX1 = 0, EX21 = 1 and put Sn = X1 + ··· + Xn. We investigate the a.s. limiting behavior of (equation presented) for general norming sequences (ak). The pointwise central limit theorem shows that LN converges a.s. to the normal distribution if ak - √k; in our paper we prove the surprising result that for suitably chosen (ak) the expression LN can converge also to non-Gaussian limits, in particular, any symmetric stable distribution is a possible limit of LN. We shall determine the class of limit distributions of LN and extend the result to the case when Xn belong to the domain of attraction of a stable law.

AB - Let X1, X2, . . . be i.i.d. r.v.'s with EX1 = 0, EX21 = 1 and put Sn = X1 + ··· + Xn. We investigate the a.s. limiting behavior of (equation presented) for general norming sequences (ak). The pointwise central limit theorem shows that LN converges a.s. to the normal distribution if ak - √k; in our paper we prove the surprising result that for suitably chosen (ak) the expression LN can converge also to non-Gaussian limits, in particular, any symmetric stable distribution is a possible limit of LN. We shall determine the class of limit distributions of LN and extend the result to the case when Xn belong to the domain of attraction of a stable law.

KW - Mixtures

KW - Pointwise central limit theorem

KW - Stable distributions

UR - http://www.scopus.com/inward/record.url?scp=0030590336&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030590336&partnerID=8YFLogxK

U2 - 10.1016/0167-7152(95)00192-1

DO - 10.1016/0167-7152(95)00192-1

M3 - Article

AN - SCOPUS:0030590336

VL - 29

SP - 361

EP - 368

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 4

ER -