Geodesic
Asymptotic behaviour of maximum of sums of i.i.d. random variables along monotone blocks
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 7, Tome 311 (2004), pp. 179-189
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Let $\{ X_i,Y_i\}_{i=1,2,\dots }$ be an i.i.d. sequence of bivariate random vectors with $P(Y_1=y)=0$ for all $y$. Put $M_n(j)=\max _{0\le k\le n-j} (X_{k+1}+\dots X_{k+j})I_{k,j},$ where $I_{k,k+j}=I\{Y_{k+1}<\dots denotes the indicator function for the event in the brackets, $1\le j\le n$. Let $L_n$ be the largest $l\le n$, for which $I_{k,k+l}=1$ for some $k=0,1,\dots,n-l$. The strong law of large numbers for “the maximal gain over the longest increasing runs”, i.e. for $M_n(L_n)$ has been recently derived for the case of $X_1$ with a finite moment of the order $3+\varepsilon,\varepsilon>0$. Assuming that $X_1$ has a finite mean we prove for any $a=0,1,\dots$, that the s.l.l.n. for $M_{(L_n-a)}$ is equivalent to ${\mathbf E}X_1^{3+a}I\{X_1>0\}<\infty$. We derive also some new results for the a.s. asymptotics of $L_n$. 
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     author = {A. I. Martikainen},
     title = {Asymptotic behaviour of maximum of sums of i.i.d. random variables along monotone blocks},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {179--189},
     year = {2004},
     volume = {311},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_311_a9/}
}
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A. I. Martikainen. Asymptotic behaviour of maximum of sums of i.i.d. random variables along monotone blocks. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 7, Tome 311 (2004), pp. 179-189. http://geodesic.mathdoc.fr/item/ZNSL_2004_311_a9/

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