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Consider a stochastic approximation process with $$x_{t+1} = x_t + \frac{1}{t} (g(x_t)+u_t)$$ where $(u_s)_s$ is a sequence of i.i.d. shocks. Assume $g$ is Lipschitz, $u_t$ has finite variance, and that $(x_s)_s$ is bounded with probability one. Furthermore, assume that $$ C = \{ x \in \mathbb{R} \colon g(x)=0\} $$ is finite.

In this case it follows from the results in stochastic approximation theory that $x$ converges a.s. to a point in $C$ (see for example Kushner and Yin, ``Stochastic Approximation and Recursive Algorithms and Applications'' Theorem 2.1 page 127).

Question: Do you know a reference that provides conditions such that $x$ converges to a given point in $C$ with strictly positive probability.

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    $\begingroup$ There seems to be an error in your definition of stable points. $\endgroup$
    – Piyush Grover
    Commented Jan 17, 2020 at 21:20
  • $\begingroup$ Thanks a lot, corrected. $\endgroup$
    – Peter
    Commented Jan 20, 2020 at 18:32
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    $\begingroup$ I still don't think it is correct. $\endgroup$
    – Piyush Grover
    Commented Jan 20, 2020 at 20:24
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    $\begingroup$ I don't understand it. The term g(x)(x-y) cannot be positive for all y close to x, since suppose $y_1>x$, then it requires g(x)<0, but then for $y_2<x$ close to x, it gives $g(x)(x-y_2)<0$. $\endgroup$
    – Piyush Grover
    Commented Jan 21, 2020 at 21:24
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    $\begingroup$ Can you point out where in the Kushner/Yin book is this definition of stability given ? $\endgroup$
    – Piyush Grover
    Commented Jan 22, 2020 at 4:20

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