On a continuous analogue of the stochastic difference equation Xn=[rho]Xn-1+Bn

Let B1, B2, ... be a sequence of independent, identically distributed random variables, letX0 be a random variable that is independent ofBn forn[greater-or-equal, slanted]1, let [rho] be a constant such that 0<[rho]<1 and letX1,X2, ... be another sequence of random variables that are defined recursively by the relationshipsXn=[rho]Xn-1+Bn. It can be shown that the sequence of random variablesX1,X2, ... converges in law to a random variableX if and only ifE[log+|B1|]<[infinity]. In this paper we let {B(t):0[less, double equals]t<[infinity]} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0[less-than-or-equals, slant]t<[infinity]} that stands in the same relationship to the stochastic process {B(t):0[less-than-or-equals, slant]t<[infinity]} as the sequence of random variablesX1,X2,...stands toB1,B2,... It is shown thatX(t) converges in law to a random variableX ast -->+[infinity] if and only ifE[log+|B(1)|]<[infinity] in which caseX 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.