Let {(X_i, Y_i)} be a sequence of independent equidistributed random, vectors with P(Y₁ = 1) = p = 1 - P(Y ₁ = 0) ∈ (0,1). Let M_n(j) = max _0≤k≤n-j(X_k+1 + ⋯ + X_k+j)I _k,j , where I_k,j = I{Y_k+1 = ⋯ = Y _k+j = 1} and I{·} denotes the indicator function of the event in brackets. If, for example, {X_i} are the gains and {Y_i} are the indicators of success in repetitions of a game of chance, then M _n(j) is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values M_n(j), j = j_n ≤ L_n, where L _n is the length of the longest head run in Y₁, . . . ,Y_n. We show that the asymptotics of the values M_n(j) depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the Erdos-Rényi law of large numbers) to the strong invariance (as in the Csörgo-Révész strong approximation laws). We also consider the Shepp-type statistics.