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On the maximal excursion over increasing runs. (English) Zbl 1069.60032

Balakrishnan, N. (ed.) et al., Asymptotic methods in probability and statistics with applications. Papers from the international conference, St. Petersburg, Russia, June 24–28, 1998. Boston, MA: Birkhäuser (ISBN 0-8176-4214-5). 225-242 (2001).
Summary: Let \(\{(X_i,Y_j)\}\) be a sequence of i.i.d. random vectors with \(P(Y_1= y)= 0\) for all \(y\). Put \(M_n(j)= \max_{0\leq k\leq n-j}(X_{k+1}+\cdots+ X_{k+j})I_{k,j}\), where \(I_{k,j}= I\{Y_{k+1}\leq\cdots\leq Y_{k+j}\}\), \(I\{\cdot\}\) denotes the indicator function of the event in brackets. If, for example, \(X_i= Y_i\), \(i\geq 1\), and \(X_i\) denotes the gain in the \(i\)th repetition of a game of chance, then \(M_n(j)\) is the maximal gain over increasing runs of length \(j\). We investigate the asymptotic behaviour of \(M_n(j)\), \(j= j_n\leq L_n\), where \(L_n\) is the length of the longest increasing run in \(Y_1,\dots, Y_n\). We show that the asymptotics of \(M_n(j)\) crucially depend on the growth rate of \(j\), and they vary from strong non-invariance like in the Erdős-Rényi law of large numbers to strong invariance like in the Csörgő-Révész strong approximation laws.
For the entire collection see [Zbl 0995.00013].

MSC:

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60F10 Large deviations
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