Tails of passage-times and an application to stochastic processes with boundary reflection in wedges

Iasnogorodski, Roudolf

doi:10.1016/s0304-4149(96)00118-4

Cited by 14 publications

(41 citation statements)

References 8 publications

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“…Lamperti [31] was the first to systematically study the problem of the existence or non-existence of moments E[η q 1 ]: his results covered only integer q. Subsequently Aspandiiarov et al extended Lamperti's results to all q > 0 (see the Appendix of [5]), but neither [31] nor the results of [5] determine whether the boundary case E[η (1+r)/2 ] is finite or infinite; as mentioned above, results of [3] can be used to settle the boundary case, but under more restrictive conditions on the increments than we use in Theorem 2.3. (The results of [5,31] related to Theorem 2.3 are stated in the Markovian setting, but their methods, similar to ours, work more generally.…”

Section: The Duration Of An Excursionmentioning

confidence: 99%

Excursions and path functionals for stochastic processes with asymptotically zero drifts

Hryniv

Menshikov

Wade

2013

Stochastic Processes and their Applications

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We study discrete-time stochastic processes (X t ) on [0, ∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form s≤t X α s , α > 0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional 'centrally biased' random walks on R d ; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al. We benefitted in the early stages of this project from enjoyable discussions with Iain MacPhee, who sadly passed away on 13th January 2012; we dedicate this paper to Iain, in memory of our valued colleague and in gratitude for his generosity.

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Section: The Duration Of An Excursionmentioning

confidence: 99%

Excursions and path functionals for stochastic processes with asymptotically zero drifts

Hryniv

Menshikov

Wade

2013

Stochastic Processes and their Applications

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show abstract

“….}. The possible transitions of the chain are specified by a 2 × 2 reinforcement matrix A = (a ij ) 2 i,j=1 and the transition probabilities depend on the current state:…”

mentioning

confidence: 99%

The simple harmonic urn

Crane¹,

Georgiou²,

Volkov³

et al. 2011

Ann. Probab.

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We study a generalized Polya urn model with two types of ball. If the drawn ball is red it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colours are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colours swap, the process is positive-recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth-death processes, a uniform renewal process, the Eulerian numbers, and Lamperti's problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally we discuss some related models of independent interest, including a "Poisson earthquakes" Markov chain on the homeomorphisms of the plane

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“…Fundamental work of Lamperti [25,26] showed that such processes are near-critical from the point of view of recurrence classification. We prove Proposition 3.1 using results from [1,2], which generalize Lamperti's work [26].…”

Section: Heavy-tailed Random Walks On Stripsmentioning

confidence: 61%

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

Hryniv

MacPhee

Menshikov

et al. 2012

Electron. J. Probab.

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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

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Tails of passage-times and an application to stochastic processes with boundary reflection in wedges

Cited by 14 publications

References 8 publications

Excursions and path functionals for stochastic processes with asymptotically zero drifts

Excursions and path functionals for stochastic processes with asymptotically zero drifts

The simple harmonic urn

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

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