Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at \bx \in \Z^d is of magnitude O(\| \bx\|^{-1}), we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude \| \bx\|^{-\beta}, $\beta \in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on $2$nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model

Strong transience for one-dimensional Markov chains with asymptotically zero drifts

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob $h$-transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.Comment: 26 pages; v2: minor revisions, expanded discussio

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes

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

The study of discrete-time stochastic processes on the half-line with mean drift at x given by [mu]1(x)-->0 as x-->[infinity] is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where [mu]1(x) is of order x-[beta] for some [beta][set membership, variant](0,1). The bounds are of order t1/(1+[beta]), 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+2[beta]+[epsilon])-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 x[beta][mu]1(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 [beta]=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.Lamperti's problem Almost-sure bounds Law of large numbers Central limit theorem Birth-and-death chain Transience Inhomogeneous random walk

Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give natural conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite. We develop and apply novel semimartingale criteria for studying explosions and establishing strong laws, which are of independent interest

Random walks avoiding their convex hull with a finite memory

Fix integers $d \geq 2$ and $k\geq d-1$. Consider a random walk $X_0, X_1, \ldots$ in $\mathbb{R}^d$ in which, given $X_0, X_1, \ldots, X_n$ ($n \geq k$), the next step $X_{n+1}$ is uniformly distributed on the unit ball centred at $X_n$, but conditioned that the line segment from $X_n$ to $X_{n+1}$ intersects the convex hull of $\{0, X_{n-k}, \ldots, X_n\}$ only at $X_n$. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-$k$ model, and comment on some open problems. In the case where $d=2$ and $k=1$, we obtain the limiting speed explicitly: it is $8/(9\pi^2)$.Comment: 31 pages, 3 figures; v2: minor revision

Reflecting random walks in curvilinear wedges

We study a random walk (Markov chain) in an unbounded planar domain bounded by two curves of the form x2=a+xβ+1 and x2=−a−xβ−1 , with x1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle α+ or α− to the relevant inwards-pointing normal vector. Here we focus on the case where α+ and α− are equal but opposite, which includes the case of normal reflection. For 0 ≤ β+, β− < 1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times