On a continuous analogue of the stochastic difference equation Xn=ρ Xn-1 + Bn

Abstract

Let B 1, B 2, ... be a sequence of independent, identically distributed random variables, let X 0 be a random variable that is independent of B n for n⩾1, let ρ be a constant such that 0<ρ<1 and let X 1, X 2, ... be another sequence of random variables that are defined recursively by the relationships X n =ρ X n-1 + B n . It can be shown that the sequence of random variables X 1, X 2, ... converges in law to a random variable X if and only if E[log +¦ B 1¦]<∞. In this paper we let { B( t):0≦ t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process { X( t):0⩽ t<∞} that stands in the same relationship to the stochastic process { B( t):0⩽ t<∞} as the sequence of random variables X 1, X 2,...stands to B 1, B 2,.... It is shown that X( t) converges in law to a random variable X as t →+∞ if and only if E[log +¦ B(1)¦]<∞ in which case X has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.