Rate of escape and central limit theorem for the supercritical Lamperti problem

Abstract

The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β(0,1). The bounds are of order t1/(1+β), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where xβμ1(x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks

Random Walk with Barycentric Self-interaction

“…In the case β ∈ [0, 1), the difference between the two settings is clearly manifest in the constant in the law of large numbers. The analogue of our Theorem 2.2 in the case of drift relative to the origin is an immediate consequence of Theorem 2.2 of [27] with Theorem 3.2 of [31] (see the discussion in [31, Section 3.2]):…”

“…Theorem 2.5. [27,31] Suppose that (A1) holds, with the modification that (2.2) holds with x = X n instead of x = X n − G n . Suppose that d ∈ N, β ∈ [0, 1), and ρ > 0.…”

“…Such non-homogeneous 'centrally biased' walks were studied by Lamperti in [21,Section 4] and [23,Section 5]; for more recent work see e.g. [13,27,31]. Considering the process of norms Z n = X n leads to a process on…”

Coloured maximal branching process

“…A related result corresponding to the case F i (x) ∼ 1 − α i x −ε with ε ∈ (0, 1) was obtained in [20 …”

“…The latter is sometimes referred to as Lamperti's problem (cf. [19], [20]) to acknowledge the contribution of Lamperti's pioneering works [9]- [11]. A generalization of the MBP from integer-valued population processes to their realvalued analogue is considered in a series of papers by Lebedev [12]- [14], see also a review of his results in [15].…”