Strong laws for the maximal gain over increasing runs

Let {(Xi,Yi)}i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y1=y)=0 for all y. Put Mn=Mn(Ln)=max0[less-than-or-equals, slant]k[less-than-or-equals, slant]n-Ln(Xk+1+...+Xk+Ln)Ik,Ln, where Ik,l=I{Yk+1[less-than-or-equals, slant]...[less-than-or-equals, slant]Yk+l} denotes the indicator function of the event in brackets, Ln is the largest l[less-than-or-equals, slant]n, for which Ik,l=1 for some k=0,1,...,n-l. If, for example, Xi=Yi, i[greater-or-equal, slanted]1, and Xi denotes the gain in the ith repetition of a game of chance, then Mn is the maximal gain over increasing runs of maximal length Ln. We derive a strong law of large numbers and a law of iterated logarithm type result for Mn.