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December, 1966 On Dvoretzky Stochastic Approximation Theorems
J. H. Venter
Ann. Math. Statist. 37(6): 1534-1544 (December, 1966). DOI: 10.1214/aoms/1177699145


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.


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J. H. Venter. "On Dvoretzky Stochastic Approximation Theorems." Ann. Math. Statist. 37 (6) 1534 - 1544, December, 1966.


Published: December, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0146.39505
MathSciNet: MR203786
Digital Object Identifier: 10.1214/aoms/1177699145

Rights: Copyright © 1966 Institute of Mathematical Statistics


Vol.37 • No. 6 • December, 1966
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