On asymptotics of the maximal gain without losses

✍ Scribed by Andrei N. Frolov

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 251 KB
• Volume: 63
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(02)00420-0

✦ Synopsis

Let {X i } be a sequence of i.i.d. random variables. Put M n = max 06k6n-j (X k+1 + • • • + X k+j )I kj , where j = j n 6 n, I kj denotes the indicator function of the event {X k+1 ¿ 0; : : : ; X k+j ¿ 0}. If X i is a gain in the ith repetition of a game of chance then M n is the maximal gain over runs without losses. We ÿnd a universal norming sequence in strong laws for M n type maxima. Our universal results yield SLLN, Erdős-Rà enyi SLLN, Cs orgő-Rà evà esz laws and LIL for such maxima. New results are obtained for distributions attracting to normal law and completely asymmetric stable laws with index ∈ (1; 2).