Vincent Granville, in his $Analytic\ Bridge$ blog posed a problem on self-correcting random walk.

Quoting from the post:

Let's start with $X(1)=0$, and define $X(k)$ recursively as follows, for $k>1$: $$ X(k) = \begin{cases} X(k-1) + \frac{U(k)}{k^a} \text{ if $X(k-1) < 0$} \\ \\ X(k-1) - \frac{U(k)}{k^a} \text{ if $X(k-1) \ge 0$} \end{cases} $$ and let's define $U(k), Z(k)$ and $Z$ as follows: \begin{align*} Z(k) &= k^a \ X(k) \\ \ \\ Z &= \lim_{k \rightarrow \infty} Z(k) \\ \ \\ U(k) &= V(k)^b \end{align*} where the $V(k)$'s are deviates from $\textit{independent}$ uniform variables on $[0,1]$ [...].

Prove that if $0<a<1$, then $X(k)$ converges to $0$ as $k$ increases. Under the same condition, prove that the limiting distribution $Z$

  • always exists,
  • always takes values between $-1$ and $+1$, with $\min(Z) = -1$, and $\max(Z) = +1$,
  • is symmetric, with mean and median equal to 0
  • and does not depend on a, but only on b.

$\{X(k)\}_k$ and $\{Z(k)\}_k$ are realizations of a non-stationary Markov process.

I've made some progress towards answering these questions and I am looking for similar problems in the literature, but I can't think where to start. Any suggestion would be appreciated.

  • $\begingroup$ Tom Salisbury provided some interesting comments. He mentioned a 1994 paper by Madras and Tanny as a possible related reference. – VictorZurkowski $\endgroup$ – VictorZurkowski Dec 17 '18 at 21:59

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