Stochastic gradient algorithm with random truncations

Let f : R-d x R-d' --> R be a Borel-measurable function which satisfies integral(Rd')\f(theta,x)\q(0)(dx) < infinity, For All theta is an element of R-d, where q(0)(.) is a probability measure on (R-d', B-d'). The problem of minimization of the function f(0)(theta) = integral(Rd') f(theta,x)q(0)(dx), theta is an element of R-d, is considered for the case when the probability measure q(0)(.), is unknown, but a realization of a non-stationary random process {X-n}(n greater than or equal to 1) whose single probability measures in a certain sense tend to q(0)(.), is available. The random process {X-n}(n greater than or equal to 1) is defined on a common probability space, R-d'-valued, correlated and satisfies certain uniform mixing conditions. The function f(.,.) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f(0)(.), and its almost sure convergence is proved. (C) 1997 Elsevier Science B.V.