Stochastic gradient algorithm with random truncations

Let ƒ : R^d × R^d′ → R be a Borel-measurable function which satisfies ∫R^d′|ƒ(θ,x)|q0(dx) < ∞, ∀θ ∈ R^d, where q0(·) is a probability measure on (R^d′,&Bauriol;^d′). The problem of minimization of the function ƒ0(θ) = ∫R^d′ƒ(θ,x)q0(dx), θ ∈ R^d, is considered for the case when the probability measure q0(·) is unknown, but a realization of a non-stationary random process {Xn}n≥1 whose single probability measures in a certain sense tend to q0(·), is available. The random process {Xn}n≥1 is defined on a common probability space, R^d′-valued, correlated and satisfies certain uniform mixing conditions. The function ƒ(·,·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of ƒ0(·), and its almost sure convergence is proved.