Oscillating random walk on Z driven by its occupation time at zero

I will consider a nearest-neighbor random walk X_n on integers with the drift at time n (conditional on the past expectation of $X_{n+1}-X_n$) equals to $-sign(X_n)f^{-1}(eta_n),$ where eta_n is the number of visits to zero by time n and f(n) is a regularly varying function with index $a geq 0.$ If a > 1, we show that the functional central limit theorem with standard normalization sqrt{n} holds for X_n. In addition, we prove that the law of the random walk is equivalent in this case to the distribution of the simple random walk. If a is in [0,1), weshow that the law of X_n/b_n converges to a non-degenerate (non-gaussian) limit distribution. Here $b_n$ is an explicit regularly varying function of index a/(1+a).

This is a joint work with Iddo Ben-Ari (UC Irvine) and Mathieu Merle (UBC).