Limiting Behavior of a Process of Runs

Abstract

Let $X_1, X_2, \cdots$ be independent identically distributed (i.i.d.) random variables with a continuous distribution function. Let $R_0 = 0, R_k = \min \{j: j > R_{k - 1}$ and $X_j > X_{j + 1}\}$ and $T_k = R_k - R_{k - 1}, k \geq 1$. We prove that a process $T^{(n)} = \{T_{k + n}\}^\infty_{k = 1}$ converges, in the sense of distribution functions, exponentially fast to a strongly mixing ergodic process. It is shown that $(\max_{1 \leq k \leq n} T_k)/\log n(\log \log n)^{-1} \rightarrow 1$ almost surely and in $L_p, p > 0$. Also, the number of runs $T_k, 1 \leq k \leq n$, larger than or equal to some $m$ is proven to be Poisson distributed in the limit, if $n/m$! converges to a positive number.