On Kersting’s Theorem on Weak Convergence of Recursions

Abstract

In an interesting paper ([1]), G. Kersting considers the asymptotic distribution of a sequence of random variables (X_n)n ∈ ℕ given by the recursion

$$ {x_{{n + 1}}} = {X_{n}} - a_{n}^{2}h\left( {{X_{n}}} \right) + {a_{n}}{Y_{n}} $$

((1))

Such types of recursive schemes are encountered in the framework of Stochastic Approximation. (Kersting derives e.g. with the help of his result the asymptotic distribution of the Robbins-Monro process when the regression function is only one-sided differentiable at the root).