Conditional limit theorems for asymptotically stable random walks

Doney, R. A.

doi:10.1007/bf00534868

Cited by 55 publications

(47 citation statements)

References 17 publications

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“…In the …rst we consider the …rst n values of a random walk conditioned on the event that all these values are non-negative; this is a discrete version of a meander, and it has been known for a long time that if the random walk is in the domain of attraction of a standard Normal law, a suitably scaled version of this process converges weakly to Brownian meander. (See [8], or [11].) The second interpretation involves conditioning on the event that the random walk never goes negative, and so can be thought of as a discrete version of the Bessel process.…”

Section: Introductionmentioning

confidence: 99%

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

Bryn-Jones¹,

Doney²

2006

J. London Math. Soc.

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Section: Introductionmentioning

confidence: 99%

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

Bryn-Jones¹,

Doney²

2006

J. London Math. Soc.

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“…For X ∈ D(2, 0) and being (h, 0) −lattice relation (9) has been obtained by Bryn-Jones and Doney [4].…”

Section: Theoremmentioning

confidence: 99%

Local probabilities for random walks conditioned to stay positive

Vatutin

Wachtel

2008

Probab. Theory Relat. Fields

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Abstract. Let S 0 = 0, {Sn, n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X 1 , X 2 , ... and let τ − = min{n ≥ 1 : Sn ≤ 0} and τ + = min{n ≥ 1 : Sn > 0}. Assuming that the distribution of X 1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as n → ∞, of the local probabilities P(τ ± = n) and the conditional local probabilities P(Sn ∈ [x, x + ∆)|τ − > n) for fixed ∆ and x = x(n) ∈ (0, ∞) .

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“…The existence of the limit in (15) is an easy consequence of the invariance principle for random walks conditioned to stay positive, which was proved by Doney [3]. The most difficult part of the proof is the derivation of characterisation (16) of the limiting distribution F α,β , see Section 3.…”

Section: 1mentioning

confidence: 96%