Abstract
We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.
Highlights
In the full-information problem, where the reward at time m is ym, such as that considered in the present study, the distribution of the observations plays a significant role in the asymptotic behaviour of the outcome statistics
We are interested in the problem where we find an optimal stopping rule that maximises our expected gain
In Example 4, we consider a broad class of distributions with exponential upper tails and show that the asymptotic behaviour of vn is fully determined for this class of distribution, subject to the assumption that vn increases without bound
Summary
There are many other variations of the no-information problem and significant work exists on the asymptotic properties of the stopping time (see, for example, [19,20,21]) It was shown in [22] that, in the secretary problem with a sequence of N observations,the asymptotic expectation and variance for the stopping time is 2N/e and 2/e − 5/e2 N 2 , respectively, where e is taken to be Euler’s constant. In the full-information problem, where the reward at time m is ym , such as that considered in the present study, the distribution of the observations plays a significant role in the asymptotic behaviour of the outcome statistics.
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