We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let @math and @math be two centered, weakly dependent random walks. We establish that @math for a non-random @math . In the classical setting, @math , it is well-known that @math . We prove that for any non-trivial @math one has @math and the exponent @math depends only on @math . Our result holds also in the continuous setting, when @math and @math are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.
This perspective was the initial motivation for analyzing the problems in this paper (more precisely the result given in Theorem ). In fact, the question arises from studies of extremal particles of a branching random walk in a time-inhomogeneous random environment. In the companion paper @cite_1 we show that the randomness of the environment has a slowing effect on the position of the maximal particle. Namely, the logarithmic correction to the speed is bigger than in the standard (time-homogenous) case, which is a consequence of ).
@cite_1
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A method is disclosed of preferentially accessing a group of positions (12 in FIG. 1) for a first type of call over a second type of call in a telephone system (FIG. 1) in which the positions (12) are served by a uniform or automatic call distributing office (1). In accordance with one aspect of the invention, selected ones of the positions (12) are marked unavailable for serving calls after the completion of a call of the second type. Thereafter the selected positions are accessed by overriding the unavailable state for a prescribed number of calls of the first type and, at the completion of the prescribed number of calls, the selected positions are again marked available for serving calls. In accordance with a second aspect of the invention, responsive to a first prescribed traffic delay condition in placing calls of the first type to the positions, ones of the positions are successively marked unavailable for serving calls as each completes a call. This occurs as long as the traffic delay state exists. Calls of the first class are placed to the positions that are marked unavailable by overriding the unavailable state. The positions are successively again made available for serving calls as each becomes idle for serving a new call responsive to a second more desirable prescribed traffic delay condition encountered in placing first type calls.