A note on upper functions for stochastic approximation

For the Robbins–Monro process (1) we study the upper functions $g(t)$ such that $\displaystyle \limsup_{t\to\infty}(X(t)-\theta)/g(t)=1$ a. s. In the case of continuous time $\xi(t)$ in (1) is the process with homogeneous independent increments; in the case of discrete time $d\xi(s)$, are i. i. d. random variables. The one-dimensional procedure (2) is considered in theorem 1, the multidimensional procedure (11) is studied in theorem 2. All results are obtained under the assumption of finiteness of moment generating function and are based on the theorems on large deviations for Markov processes [10].