An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn=β(ZnÅ-1)Xn+(1-β(ZnÅ-1))ZnÅ-1, where Xn are i.i.d. taking values in the interval [0,M], M••, and β is a monotonically increasing function [0,M]⇒[c,d], 0<c<d<1. Together with (Zn), we consider the ordinary exponential smoothing Yn=αXn+(1-α)YnÅ-1, where α is a constant, 0<α<1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY,EZ, respectively, with a geometric convergence rate. Moreover, we show that EZ is strictly less than EY=EXn.
