On Dvoretzky Stochastic Approximation Theorems

Let $H$ be a set and $\{T_n, n = 1, 2, \cdots\}$ a sequence of transformations of $H$ into itself. Let $X_1$ and $\{U_n\}$ be random elements in $H$ and generate the sequence $\{X_n\}$ by $X_{n + 1} = T_n(X_n) + U_n.$ Theorems giving conditions under which $\{X_n\}$ is "stochastically attracted" towards a given subset of $H$ and will eventually be within or arbitrarily close to this set in an appropriate sense, are called Dvoretzky stochastic approximation theorems. The main results of this paper (Theorems 1, 2 and 3) are of this type. They generalize the work of Dvoretzky [6] and Wolfowitz [12] for the case $H$ equal to the real line, of Derman and Sacks [4] for the case $H$ a finite dimensional Euclidian space and Schmetterer [11] for the case $H$ a Hilbert space.