Small drift limit theorems for random walks

Abstract

Abstract. We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞, 0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞, 0) we consider parametrized classes of random walks, where the convergence of the parameter to zero implies the convergence of the drift to zero. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a > 0 and then letting a tend to zero. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below zero of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulas we derive the generating function of the occupation time in closed form, which provides an alternative approach. In the course also give a new form of the first arcsine law for the Brownian motion with drift.

Independent factorization of the last zero arcsine law for Bessel processes with drift

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We show that the last zero before time t of a recurrent Bessel process with drift starting at 0 has the same distribution as the product of a right-censored exponential random variable and an independent beta random variable. This extends a recent result of Schulte-Geers and Stadje [19] from Brownian motion with drift to recurrent Bessel processes with drift. We give two proofs, one of which is intuitive, direct, and avoids heavy computations.

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“…x to denote the law of Brownian motion with drift µ when it starts at x ∈ R. Using a random walk approximation argument, Schulte-Geers and Stadje [19,Theorem 2.1] show that g t under W µ 0 has the independent factorization…”

“…Despite the elegant and elementary nature of (1.3), it seems to have escaped notice until [19]; see also [4,Remark 2.1]. The present author learned of this result from a MathOverflow answer [3] posted by the first author of [19]. Another attractive feature of (1.3) is that it allows us to easily recover Lévy's last zero arcsine law (1.1) since min{t, E µ } degenerates to t as µ → 0.…”

“…Another derivation which uses a formula from Borodin and Salminen's handbook [2] also requires significant computations; see Section 4.2. Indeed, in [19,Remark 2.3], the authors appeal for a "purely Brownian" explanation of the independent factorization (1.3). This leads us to the main contributions of this paper:…”

Independent factorization of the last zero arcsine law for Bessel processes with drift

“…Despite the elegant and elementary nature of (3), it seems to have escaped notice until [SGS17], see also [IO20,Remark 2.1]. The present author first learned of this characterization from a MathOverflow answer [esg], presumably written by the first author of [SGS17].…”