On the range of self-interacting random walks on an integer interval

Abstract

We consider the range of a one-parameter family of self-interacting walks on the integers up to the time of exit from an interval. We derive the weak convergence of an appropriately scaled range. We show that the distribution functions of the limits of the scaled range satisfy a certain class of de Rham's functional equations. We examine the regularity of the limits. * AMS 2000 subject classifications : 60K35. Key words and phrases : self-interacting random walk, range of random walk.

On regularity for de Rham’s functional equations

“…See Example 2.2 (iii) for details. Our main results are also applicable to some examples which are even outside the framework of [10] and [11]. In Example 2.2 (i) (resp.…”

“…Our main results are applicable to some cases to which the results in [3,Section 7] are not applicable. [3, (7.3) in Theorem 7.3] corresponds to m−1 i=0 max x∈X Df i (x) < m −m , and, [3, (7.6) [10] and [11], the author considered some regularities of G if X = [0, 1], m = 2, and, f i s are certain linear fractional transformations on X. max Df 0 max Df 1 ≥ 1/4 ≥ min Df 0 min Df 1 can occur. By [14], the inverse function of Minkowski's question-mark function is the solution of (1.1) for the case that X = [0, 1], m = 2, f 0 (x) := x/(x+1) and f 1 (x) := 1/(2−x).…”

“…Then, max Df 0 max Df 1 = 1 and min Df 0 min Df 1 = 1/4. This case is in the framework of [11]. See Example 2.2 (iii) for details.…”

“…However, we can apply the technique in the proof of the author [11, Lemma 3.5], because 0 is not a fixed point of x → x + 1 or x → −1/(2 + x). The approach in[11, Lemma 3.5] is different from the one in this paper. [10, (A3) in Section 1] assures a stronger result than singularity, specifically, the Hausdorff dimension of µ G , dim H (µ G ) is strictly smaller than 1.…”

Singularity Results for Functional Equations Driven by Linear Fractional Transformations

“…The following example concerns the range of self-interacting walks on an interval in the author [8]. Let 0 < u < √ 3.…”