Local probabilities for random walks conditioned to stay positive

Vatutin, Vladimir; Wachtel, Vitali

doi:10.1007/s00440-007-0124-8

Cited by 84 publications

(103 citation statements)

References 27 publications

(45 reference statements)

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“…Lemma 2 (compare with Lemma 13 in [21] and Corollary 8 in [7]) If X ∈ D (α, β) , then there exists a function l 0 (x) slowly varying at infinity such that…”

Section: Auxiliary Resultsmentioning

confidence: 99%

How Many Families Survive for a Long Time?

Vatutin¹,

Dyakonova²

2017

Theory Probab. Appl.

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A critical branching process {Z k , k = 0, 1, 2, ...} in a random environment generated by a sequence of independent and identically distributed random reproduction laws is considered. Let Zp,n be the number of particles at time p ≤ n having a positive offspring number at time n. A theorem is proved describing the limiting behavior, as n → ∞ of the distribution of a properly scaled process log Zp,n under the assumptions Zn > 0 and p ≪ n.

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“…Lemma 2 (compare with Lemma 13 in [21] and Corollary 8 in [7]) If X ∈ D (α, β) , then there exists a function l 0 (x) slowly varying at infinity such that…”

Section: Auxiliary Resultsmentioning

confidence: 99%

How Many Families Survive for a Long Time?

Vatutin¹,

Dyakonova²

2017

Theory Probab. Appl.

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“…This statement is contained implicitly in Lemma 13 of [11]. In our proofs we shall frequently use the following well-known properties of regularly varying sequences.…”

Section: Remark 6 Representationmentioning

confidence: 99%

Local limit theorem for the maximum of asymptotically stable random walks

Wachtel¹

2010

Probab. Theory Relat. Fields

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Let {S n ; n ≥ 0} be an asymptotically stable random walk and let M n denote it's maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal massfunction based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M n does not hold.

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“…, G n v ∈ B c } and find the limit law for the quantity 1 √ n log G n v conditioned that τ v > n. The study of related problems for random walks in R has attracted much attention. We refer the reader to Spitzer [28], Iglehart [20], Bolthausen [5], Bertoin and Doney [2], Doney [10], Borovkov [3,4], Vatutin and Wachtel [30], Caravenna [7] and to references therein. Random walks in R d conditioned to stay in a cone have been considered in Shimura [27], Garbit [14], Echelsbacher and König [11] and Denisov and Wachtel [8,9].…”

Section: Introductionmentioning

confidence: 99%

Conditioned limit theorems for products of random matrices

Grama

Page

Peigné

2016

Probab. Theory Relat. Fields

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Let g 1 , g 2 , . . . be i.i.d. random matrices in G L (d, R) . For any n ≥ 1 consider the product G n = g n . . . g 1 and the random processIt is well known that under appropriate assumptions, the sequence (log G n v ) n≥1 behaves like a sum of i.i.d. r.v.'s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with v > 1 denote by τ v the first time when the random process G n v enters the closed unit ball in R d . We establish the asymptotic as n → +∞ of the probability of the event {τ v > n} and find the limit law for the quantity 1 √ n log G n v conditioned that τ v > n.

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Local probabilities for random walks conditioned to stay positive

Cited by 84 publications

References 27 publications

How Many Families Survive for a Long Time?

How Many Families Survive for a Long Time?

Local limit theorem for the maximum of asymptotically stable random walks

Conditioned limit theorems for products of random matrices

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