# Self-correcting Random Walks

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 $$0, then $$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.

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