A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

Bryn-Jones, A.; Doney, R. A.

doi:10.1112/s0024610706022964

Cited by 31 publications

(40 citation statements)

References 19 publications

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“…The present work rests on the results for the case A = {0} given in [12] for which the harmonic analysis is effectively applicable but does not seem to be for the non-lattice walks. However, another approach seems promising to lead to corresponding results at least under some assumption on the distribution of the increment variable: one may take the half line (−∞, 0] in place of {0} (see Remark 4 of Section 5) and apply the corresponding results for the transition probability that are found in several papers [1], [2], [4], [16], etc. and those for the potential operator given in [9].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

One dimensional random walks killed on a finite set

Uchiyama

2017

Stochastic Processes and their Applications

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Section: Introduction and Main Resultsmentioning

confidence: 99%

One dimensional random walks killed on a finite set

Uchiyama

2017

Stochastic Processes and their Applications

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“…We point out that Theorem 4 has been obtained also in [4], where the authors study random walks conditioned to stay positive in a different sense.…”

Section: Theoremmentioning

confidence: 92%

A local limit theorem for random walks conditioned to stay positive

Caravenna

2005

Probab. Theory Relat. Fields

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We consider a real random walk S n = X 1 +. . .+X n attracted (without centering) to the normal law: this means that for a suitable norming sequence a n we have the weak convergence S n /a n ⇒ ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let C n denote the event (S 1 > 0, . . . , S n > 0) and let S + n denote the random variable S n conditioned on C n : it is known that S + n /a n ⇒ ϕ + (x)dx, where ϕ + (x) := x exp(−x 2 /2)1 (x≥0) . What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X 1 has an absolutely continuous law: in this case the uniform convergence of the density of S + n /a n towards ϕ + (x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so-called Fluctuation Theory for random walks.

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“…Some asymptotic estimates of the transition probability of the walk killed on a half line are obtained in [13], [14], [15], [16] and [12]. In the first four papers the problem is considered for wider classes of random walks: the variance may be infinite with the law of Y in the domain of attraction of a stable law [16], [15] or of the normal law [13] [14]; Y is not necessarily restricted to the arithmetic variables. The very recent paper [15] describes the asymptotic behavior of p n (−∞,0] (x, y) valid uniformly within the region of stable deviation of the space-time variables and it in particular contains Theorem 1.3 and Corollary 1.1 as a special case.…”

Section: Proposition 61mentioning

confidence: 99%

One dimensional lattice random walks with absorption at a point/on a half line

Uchiyama¹

2011

J. Math. Soc. Japan

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This paper concerns a random walk that moves on the integer lattice and has zero mean and a finite variance. We obtain first an asymptotic estimate of the transition probability of the walk absorbed at the origin, and then, using the obtained estimate, that of the walk absorbed on a half line. The latter is used to evaluate the space-time distribution for the first entrance of the walk into the half line. 1 1 key words: absorption, transition probability, asymptotic estimate, one dimensional random walk

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A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

Cited by 31 publications

References 19 publications

One dimensional random walks killed on a finite set

One dimensional random walks killed on a finite set

A local limit theorem for random walks conditioned to stay positive

One dimensional lattice random walks with absorption at a point/on a half line

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