Some Invariant Laws Related to the Arc Sine Law

Abstract

Let $\{X_i : i = 1, 2, \cdots\}$ be a sequence of random variables such that $X_1, \cdots, X_n$ are exchangeable and symmetric $(n = 1, 2, \cdots)$. Suppose that ties occur with probability zero among the partial sums $S_0 = 0, S_k = \sum^k_1X_i$. We study the laws of the variables $J_k$, the index of the $k$th positive sum in the sequence $S_1, S_2, \cdots (k = 1, 2, \cdots), N'_n$, the number of positive sums among $S_0, S_1, \cdots, S_{L_n}$, where $L_n$ is the index of $\max \{S_0, S_1, \cdots, S_n\}$. Brief attention is given to $J_k$ in [2], where the simple form of its law in the symmetric case is however not mentioned. The variable $N_n'$ does not seem to have been considered before. Setting $a_k = 2^{-2k}\binom{2k}{k},\quad k = 0, 1, 2, \cdots (a_0 = 1),$ we find the probabilities \begin{equation*}\tag{1.1}q_k(n) = P\lbrack J_k = n\rbrack = (k/n)a_ka_{n-k},\quad n = k, k + 1, \cdots,\end{equation*} \begin{equation*}\tag{1.2}p_n(i) = P\lbrack N_n' = i\rbrack = (2ia_i)^{-1}a_n,\quad i = 1, 2, \cdots, n (p_n(0) = a_n).\end{equation*} Let $\{X_t, 0 \leqq t \leqq T < \infty\}$ be a measurable, separable stochastic process which is continuous in probability and has exchangeable, symmetric increments. Relative to the bounded time interval $0 \leqq t \leqq T$, introduce the variables (1.3) \begin{align*}U = \text{"time spent in the positive half plane up to the moment when the process reaches its maximum,"} \\ V = \text {"time elapsed until the process reaches its maximum."}\end{align*} Asymptotic evaluations lead to THEOREM. $U/V$ is independent of $V/T$, and for $0 \leqq \alpha, \gamma \leqq 1$, $P\lbrack U < \alpha V\rbrack = 1 - (1 - \alpha)^{\frac{1}{2}},\quad P\lbrack U < \gamma T\rbrack = \gamma^{\frac{1}{2}}.$