On asymptotic behaviour of increments of sums over head runs

Let $\{(X_i,Y_i)\}$ be a sequence of i.i.d. random vectors with $P(Y_1=1)=p=1-P(Y_1=0)\in (0,1)$. Put $M_n(j)=\max_{0\le k\le n-j}(X_{k+1}+\dots+X_{k+j})I_{k,j}$, where $I_{k,j}=I\{Y_{k+1}=\dots=Y_{k+j}=1\}$, $I\{\,\cdot\,\}$ denotes the indicator function of the event in brackets. If, for example, $\{X_i\}$ are gains and $\{Y_i\}$ are indicators of successes in repetitions of a game of chance, then $M_n(j)$ is the maximal gain over head runs (sequences of successes without interruptions) of length $j$. We investigate the asymptotic behaviour of $M_n(j)$, $j=j_n\le L_n$, where $L_n$ is the length of the longest head 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. We also consider Shepp type tatistics.