THE ASYMPTOTIC-BEHAVIOR OF THE REWARD SEQUENCE IN THE OPTIMAL STOPPING OF IID RANDOM-VARIABLES

Let X1, X2,... be integrable, i.i.d. r.v.'s with common distribution function F and let {upsilon-n}n greater-than-or-equal-to 1 be the sequence of optimal rewards or values in the associated optimal stopping problem, i.e., upsilon-n = sup{E(X(T)): T is a stopping time for {X(m)}m greater-than-or-equal-to 1 and T less-than-or-equal-to n} for n greater-than-or-equal-to 1. For distribution functions F in the domain of attraction of one of the three classical extreme-value laws G(I), G(II)alpha or G(III)alpha, it is shown that lim(n) n(1 - F(upsilon-n)) = 1, 1- alpha-1, or 1 + alpha-1 if F is-a-member-of D(G(I)), F is-a-member-of D(G(II)alpha) and alpha > 1, or F is-a-member-of D(G(III)alpha) and alpha > 0, respectively. From this result, the growth rate of {upsilon}n greater-than-or-equal-to 1 is obtained and compared to the growth rate of the expected maximum sequence. Also, the limit distribution of the optimal reward r.v.'s {X(Tn)*}n greater-than-or-equal-to 1 is derived, where {T(n)*}n greater-than-or-equal-to 1 are the optimal stopping times defined by T(n)* = 1 if n = 1 and, for n = 2,3,..., by T(n)* = min{1 less-than-or-equal-to k < n: X(k) > upsilon-n - k} if this set is not equal to phi and equal to n otherwise. This tail-distribution growth rate is shown to be sufficient for any threshold sequence to be asymptotically optimal.