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On asymptotics of the maximal gain without losses

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  • Frolov, Andrei N.

Abstract

Let {Xi} be a sequence of i.i.d. random variables. Put Mn=max0[less-than-or-equals, slant]k[less-than-or-equals, slant]n-j(Xk+1+...+Xk+j)Ikj, where j=jn[less-than-or-equals, slant]n, Ikj denotes the indicator function of the event {Xk+1[greater-or-equal, slanted]0,...,Xk+j[greater-or-equal, slanted]0}. If Xi is a gain in the ith repetition of a game of chance then Mn is the maximal gain over runs without losses. We find a universal norming sequence in strong laws for Mn type maxima. Our universal results yield SLLN, Erdös-Rényi SLLN, Csörgo-Révész laws and LIL for such maxima. New results are obtained for distributions attracting to normal law and completely asymmetric stable laws with index [alpha][set membership, variant](1,2).

Suggested Citation

  • Frolov, Andrei N., 2003. "On asymptotics of the maximal gain without losses," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 13-23, May.
    • Handle: RePEc:eee:stapro:v:63:y:2003:i:1:p:13-23
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    References listed on IDEAS

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    1. Frolov, Andrei N., 1998. "On one-sided strong laws for large increments of sums," Statistics & Probability Letters, Elsevier, vol. 37(2), pages 155-165, February.
    2. Frolov, Andrei N. & Martikainen, Alexander I., 1999. "On the length of the longest increasing run in d," Statistics & Probability Letters, Elsevier, vol. 41(2), pages 153-161, January.
    3. Frolov, Andrei & Martikainen, Alexander & Steinebach, Josef, 2000. "Strong laws for the maximal gain over increasing runs," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 305-312, November.
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