On asymptotic behavior of increments of sums along head runs (Q1603213)

The author investigates the almost sure limiting behaviour of the maxima along head runs, i.e. of \[ M_n(j) = \max_{0\leq k\leq n-j} (S_{k+[j]} - S_k) I \{ Y_{k+1} = \ldots = Y_{k+[j]} = 1 \} \;, \] where \(\{ (X_n ,Y_n)\}\) is a sequence of iid random vectors with \(P(Y_1=1)= p = 1-(P(Y_1)=0) \in (0,1]\), and \(S_n = X_1 + \ldots + X_n\), \(S_0 =0\). The results depend significantly on the growth rate of \(j=j_n\), and they vary from strong invariance (for large increments) to strong noninvariance (for small increments). It turns out that the corresponding ranges of \(j=j_n\) are different from their counterparts for maxima over increasing runs, i.e. for \[ M_n'(j) = \max_{0\leq k \leq n-j} (S_{k+[j]} - S_k) I \{ Y_{k+1} \leq \ldots \leq Y_{k-j}\} , \] for which similar results have been established by the author, \textit{A. Martikainen} and the reviewer [Stat. Probab. Lett. 50, No. 3, 305-312 (2000; Zbl 0967.60026)].