Some Joint Laws in Fluctuation Theory

Abstract

Let $S_0 = 0, S_n = \sum^n_1 X_i, n = 1,2, \cdots$ be the sequence of partial sums of independent, identically distributed real valued random variables $X_1, X_2, \cdots$ The sum $S_k$ is a (strict) ladder sum if $S_j < S_k$ for $0 \leqq j < k$. Contrary to usual practice, we always count $S_0$ as a ladder sum. A run of ladder sums of length $i$ starts at $S_k$ if $S_k, S_{k+1}, \cdots, S_{k+i-1}$ are ladder sums but $S_{k-1}$ and $S_{k+i}$ are not. Thus $S_0$ is always the beginning of a run, of length one if $S_1 \leqq 0$. Except at the beginning of Section 3, we assume that $X_1$ has continuous law, symmetric with respect to 0. Thus ties $S_i = S_j$ can be disregarded. Let \begin{align*} L_n &= "\text{index of} \max \{S_i, 0 \leqq i \leqq n\}," \\ \tag{1} G_n &= "\text{number of ladder sums among} 0, S_1, \cdots, S_n," \\ R_n &= "\text{number of runs they form}." \\ \end{align*} Two sets of probabilities concerning those variables are obtained, namely the joint law \begin{equation*}\tag{2} p_n(k + 1, m + 1) = P(G_n = k + 1, R_n = m + 1) = 2^{-2n+k}\binom{2n - 2k}{n - k - m}\binom{k}{m}\end{equation*} where $0 \leqq m \leqq k \leqq n$ and $k + m \leqq n$, and the probabilities \begin{equation*}\tag{3} p_n^\ast(k + 1, m + 1) = P(G_n = k + 1, R_n = m + 1, L_n = n) = \frac{m}{n - k}p_n(k + 1, m + 1),\end{equation*} valid for $1 \leqq m \leqq k \leqq n$ and $k + m \leqq n$. If $m = 0$ and $0 \leqq k < n$, then clearly $p_n^\ast(k + 1, 1) = 0$, while on the other hand $p_n^\ast(n + 1, 1) = P(0 < S_1 < \cdots < S_n) = 2^{-n}$. Considering similarly the behavior beyond the maximum, let $X_i' = -X_{n-i+1}, S'_i = \sum^i_{k=1} X'_k = S_{n-1} - S_n, i = 1, \cdots, n$, and \begin{align*}\tag{4} G'_n &= "\text{number of ladder sums among} 0, S'_1, \cdots, S'_n," \\ R'_n &= "\text{number of runs they form}."\end{align*} so that $G'_n$ counts descending ladders from $S_{L_n}$ on. Various limit laws are obtained, in particular it is shown that $(2L_n)^{-\frac{1}{2}} G_n, (2n - 2L_n)^{-\frac{1}{2}}G'_n, n^{-1} L_n, G^{-\frac{1}{2}}_n (2R_n - G_n)$ and $G_n^{'- \frac{1}{2}}(2R'_n - G'_n)$ are asymptotically independent, the first two having a limiting $\chi_2$ law and the latter two a standard normal one.