Let {(X_i,Y_i)}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁=y)=0 for all y. Put M_n=M_n(L_n)=max_0≤k≤n-Ln(X _k+1++X_k+Ln)I_k,Ln, where I_k,ℓ=I{Y_k+1≤≤Y_k+ℓ} denotes the indicator function of the event in brackets, L_n is the largest ℓ≤n, for which I_k,ℓ=1 for some k=0,1,...,n-ℓ. If, for example, X_i=Y_i, i≥1, and X_i denotes the gain in the ith repetition of a game of chance, then M_n is the maximal gain over increasing runs of maximal length L_n. We derive a strong law of large numbers and a law of iterated logarithm type result for M_n.