On asymptotics of the maximal gain without losses

Let {Xi} be a sequence of i.i.d. random variables. Put Mn=max0[less-than-or-equals, slant]k[less-than-or-equals, slant]n-j(Xk+1+...+Xk+j)Ikj, where j=jn[less-than-or-equals, slant]n, Ikj denotes the indicator function of the event {Xk+1[greater-or-equal, slanted]0,...,Xk+j[greater-or-equal, slanted]0}. If Xi is a gain in the ith repetition of a game of chance then Mn is the maximal gain over runs without losses. We find a universal norming sequence in strong laws for Mn type maxima. Our universal results yield SLLN, Erdös-Rényi SLLN, Csörgo-Révész laws and LIL for such maxima. New results are obtained for distributions attracting to normal law and completely asymmetric stable laws with index [alpha][set membership, variant](1,2).