The Structure of the Optimal Stopping Rule in the $S_n/n$ Problem

Abstract

The proof of the existence of the optimal stopping rule for $S_n/n$ for the case where the i.i.d. random variables $X_i$ have a moment of order greater than one has been obtained by B. Davis. In the present paper the asymptotic growth of the boundary of the optimal stopping region is studied. The method used generalizes one of Shepp (1969), and involves comparison with the corresponding problem for an infinitely divisible process, obtained as a limit of processes $(S_{\lbrack nt\rbrack}/a_n, t \geqq 0)$ for properly chosen norming constants $a_n$. When the $X_i$ are in the domain of attraction of a random variable which is stable with exponent greater than one, an explicit asymptotic expression for the curve defining the boundary is obtained.