# On Asymptotics of Optimal Stopping Times

## TL;DR

This paper derives asymptotic formulas for the expectation and variance of optimal stopping times in full information problems involving independent continuous random variables, highlighting different behaviors for bounded and unbounded distributions.

## Contribution

It provides explicit asymptotic expressions for optimal stopping times and their variance, depending on distribution tail behavior, supported by calculations and simulations.

## Key findings

- Asymptotic expressions depend on distribution tail decay rate.
- Behavior differs for finite and infinite upper bounds.
- Numerical simulations confirm theoretical predictions.

## 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; this is often known as the "full information" problem. In this analysis, we obtain asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables becomes 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 these statistics depend on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions, which are compared to numerical simulations that support the asymptotic predictions.