### Abstract

Let X_{1}, X_{2}, . . . be independent, identically distributed random variables with EX_{1} = 0, EX^{2}_{1} = 1 and let S_{n} = ∑_{k ≤ n} X_{k}. We give nearly optimal criteria for an (unbounded) measurable function f to satisfy the a.s. central limit theorem, i.e., (equation presented) where φ is the standard normal density function.

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

Pages (from-to) | 67-76 |

Number of pages | 10 |

Journal | Statistics and Probability Letters |

Volume | 37 |

Issue number | 1 |

Publication status | Published - Jan 15 1998 |

### Fingerprint

### Keywords

- A.s. central limit theorem
- Logarithmic average
- Wiener process

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

*Statistics and Probability Letters*,

*37*(1), 67-76.

**Almost sure central limit theorems under minimal conditions.** / Berkes, István; Csáki, E.; Horváth, Lajos.

Research output: Contribution to journal › Article

*Statistics and Probability Letters*, vol. 37, no. 1, pp. 67-76.

}

TY - JOUR

T1 - Almost sure central limit theorems under minimal conditions

AU - Berkes, István

AU - Csáki, E.

AU - Horváth, Lajos

PY - 1998/1/15

Y1 - 1998/1/15

N2 - Let X1, X2, . . . be independent, identically distributed random variables with EX1 = 0, EX21 = 1 and let Sn = ∑k ≤ n Xk. We give nearly optimal criteria for an (unbounded) measurable function f to satisfy the a.s. central limit theorem, i.e., (equation presented) where φ is the standard normal density function.

AB - Let X1, X2, . . . be independent, identically distributed random variables with EX1 = 0, EX21 = 1 and let Sn = ∑k ≤ n Xk. We give nearly optimal criteria for an (unbounded) measurable function f to satisfy the a.s. central limit theorem, i.e., (equation presented) where φ is the standard normal density function.

KW - A.s. central limit theorem

KW - Logarithmic average

KW - Wiener process

UR - http://www.scopus.com/inward/record.url?scp=0032518006&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032518006&partnerID=8YFLogxK

M3 - Article

VL - 37

SP - 67

EP - 76

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 1

ER -