On a Fluctuation Theorem for Processes with Independent Increments II

Abstract

Let $\xi(t), t \geqq 0, \xi(0) = 0$ be a separable stochastic process with stationary independent increments whose sample functions are continuous on the right. Write $\bar{\xi}(t) = \sup_{0\leqq s\leqq t} \xi(s)$ and $T_t = \min \lbrack u:0 \leqq u \leqq t; \xi(u) = \bar{\xi}(t)\rbrack.$ The object of this paper is to establish the following theorem: THEOREM. The limiting distribution \begin{equation*}\tag{1}\lim_{t\rightarrow\infty} \operatorname{Pr}(t^{-1} T_t < x) = F(x)\end{equation*} exists if and only if \begin{equation*}\tag{2}\lim_{t\rightarrow\infty} t^{-1} \int^t_0 \operatorname{Pr}(\xi(u) > 0) du = \alpha,\quad 0 \leqq \alpha \leqq 1,\end{equation*} and then $F(x)$ is related to $\alpha$ by \begin{equation*}\begin{align*}\tag{3}F(x) &= F_\alpha(x) = (\sin \pi\alpha) \pi^{-1} \int^x_0 \nu^{-(1-\alpha)} (1 - \nu)^{-\alpha} dv, 0 < \alpha < 1, 0 \leqq x \leqq 1; \\ F_0(x) &= 0 \text{if} x < 0, 1 \text{if} x \geqq 0; \\ F_1(x) &= 0 \text{if} x < 1, 1 \text{if} x \geqq 1\end{align*} .\end{equation*} This theorem is the exact counterpart to a theorem of Spitzer ([4], Theorem 7.1) for sums of independent and identically distributed random variables. It contains as a special case the well-known arc-sine limit theorem for the Brownian motion process. An earlier version of the theorem was obtained by Heyde [3] under the additional condition $\int^1_0 t^{-1} \operatorname{Pr}(\xi(t) > 0) dt < \infty$, the violation of which leads to $\operatorname{Pr}(\bar{\xi}(t) = 0) = 0, t > 0$. It should be remarked that $T_t$ has the same distribution as $N_t = \mu\{u:0 \leqq u \leqq t; \xi(u) > 0\}, \mu$ denoting ordinary Lebesgue measure. This follows from the well-known corresponding result of Sparre-Andersen for sums of independent and identically distributed random variables by a straightforward limiting argument.