On Asymptotic Behavior of Increments of Sums Along Head Runs

Abstract

Let $$\left\{ {(\user1{X}_\user1{i} \user1{,Y}_\user1{i} )} \right\}$$ be a sequence of independent equidistributed random vectors with $$\user1{P}(\user1{Y}_1 = 1) = p = 1 - \user1{P}(\user1{Y}_1 = 0) \in (0,1)$$ . Let $$M_n (\user1{j}) = max_{0 \leqslant \user1{k} \leqslant \user1{n - j}} \left( {\user1{X}_{\user1{k} + 1} + \cdots + \user1{X}_{\user1{k} + j} } \right)\user1{I}_{\user1{k,j}} $$ , where $$\begin{gathered} \user1{I}_{\user1{k,j}} = \user1{I}\left\{ {\user1{Y}_{\user1{k} + 1} = \cdots = \user1{Y}_{\user1{k} + j} = 1} \right\} \hfill \\ (\underline s \hfill \\ \end{gathered} $$ and $$\user1{I}\{ \cdot \} $$ denotes the indicator function of the event in brackets. If, for example, $${\left\{ {\user1{X}_\user1{i} } \right\}}$$ are the gains and $${\left\{ {\user1{X}_\user1{i} } \right\}}$$ are the indicators of success in repetitions of a game of chance, then $$M_n (\user1{j})$$ is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values $$M_n (\user1{j})$$ , $$\user1{j} = \user1{j}_\user1{n} \leqslant \user1{L}_\user1{n} $$ , where $$\user1{L}_\user1{n} $$ is the length of the longest head run in $$\user1{Y}_1 \user1{,} \ldots \user1{,Y}_\user1{n} $$ . We show that the asymptotics of the values $$M_n (\user1{j})$$ depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the Erdős―Reenyi law of large numbers) to the strong invariance (as in the Csoorgő―Revesz strong approximation laws). We also consider the Shepp-type statistics. Bibliography: 17 titles.