On strong versions of the central limit theorem

✍ Scribed by Zdzisław Rychlik; Konrad S. Szuster

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 161 KB
• Volume: 61
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(02)00362-0

✦ Synopsis

The purpose of this paper is to present strong versions of the central limit theorem for independent nonidentically distributed random variables. Let S n , n ¿ 1, be the partial sums of independent random variables with zero means and ÿnite variances and let a(x) be a real function. We present su cient conditions under which the arithmetic means of a(S n k =(ES 2 n k ) 1=2 ) converge almost surely to ∞ -∞ a(x) d (x) for some subsequences {n k ; k ¿ 1}, where denotes the standard normal distribution.