Frolov, Andrei; Martikainen, Alexander; Steinebach, Josef 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]. Cited in 2 Documents MSC: 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60F10 Large deviations Keywords:head run; monotone block; increments of partial sums; Erdős-Rényi law; strong approximation; strong limit theorem; large deviations PDFBibTeX XMLCite \textit{A. Frolov} et al., in: Asymptotic methods in probability and statistics with applications. Papers from the international conference, St. Petersburg, Russia, June 24--28, 1998. Boston, MA: Birkhäuser. 225--242 (2001; Zbl 1069.60032)