A renewal theorem and supremum of a perturbed random walk

Damek, Ewa; Kołodziejek, Bartosz

doi:10.1214/18-ecp184

Cited by 3 publications

(9 citation statements)

References 18 publications

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“…approximating 1 (1,∞) (x). Observe that when g(x) is replaced by 1 (1,∞) (x) we obtain have e α(x−z) E1 (1,∞) (e z−x B) = L(e x−z ) and so Theorem 3.1 is in analogy to Theorems 3.1 and 3.3 in [15] which say that…”

Section: Renewal Theoremmentioning

confidence: 91%

“…so that ψ 0 (x) = I 1 (x) − I 2 (x) − I 3 (x). In the proof of Theorem 4.2 in [15] we have already shown (under weaker assumptions) that…”

Section: 2mentioning

confidence: 96%

“…see [15], where L(x) = x α P(B > x) is assumed to be slowly varying function, ∼ is defined in (7) and ρ is as in (A-2) 2 . More precisely, if conditions (A-1), (A-2), (B-1), (AB-1) defined in Theorem 1.1 hold then (5) follows and the right hand side of ( 5) is due to both the behavior of B and of an appropriate renewal measure determined by A.…”

mentioning

confidence: 99%

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Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

Damek

Kołodziejek

2020

Stochastic Processes and their Applications

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We study the stochastic recursion X n = Ψ n (X n−1 ), where (Ψ n ) n≥1 is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation x → Ax + B. We describe the tail behavior of the stationary solution X under the assumption that there exists α > 0 such that E|A| α = 1 and the tail of B is regularly varying with index −α < 0. We also find the second order asymptotics of the tail of X when Ψ(x) = Ax + B.

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Section: Renewal Theoremmentioning

confidence: 91%

“…so that ψ 0 (x) = I 1 (x) − I 2 (x) − I 3 (x). In the proof of Theorem 4.2 in [15] we have already shown (under weaker assumptions) that…”

Section: 2mentioning

confidence: 96%

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Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

Damek

Kołodziejek

2020

Stochastic Processes and their Applications

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“…It states that the distribution of R∞* has a heavy right tail under the assumptions A > 0 a.s., EAs=1 for some s > 0 and some additional conditions, see formula (7.39) below for more details in the particular case ( A , B ) = ( ρ , ξ ). The tail behavior of R∞* is also well understood in some other cases, in particular, when ℙ{|B|>x} is regularly varying at ∞ (see, for instance, [18], [20] and [8]).…”

Section: Branching Processes In Random Environment With Immigrationmentioning

confidence: 99%

Random walks in a moderately sparse random environment

Buraczewski

Dyszewski

Iksanov

et al. 2019

Electron. J. Probab.

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A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk (Xn)n∈ℕ∪{0} in a sparse random environment (Sk,λk)k∈ℤ is a nearest neighbor random walk on ℤ that jumps to the left or to the right with probability 1/2 from every point of ℤ\{…, S−1, S0=0, S1,…} and jumps to the right (left) with the random probability λk+1 (1 − λk+1) from the point Sk, k∈ℤ. Assuming that (Sk−Sk−1,λk)k∈ℤ are independent copies of a random vector (ξ, λ)∈ℕ×(0, 1) and the mean Eξ is finite (moderate sparsity) we obtain stable limit laws for Xn, properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case Eξ=∞ (strong sparsity) will be analyzed in a forthcoming paper.

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“…A major part of the recent book [23] is concerned with the so defined perturbed random walks, both multiplicative and additive. We refer to the cited book for numerous applications of these random sequences and to [1,15,17,24,25] for more recent contributions.…”

Section: Introductionmentioning

confidence: 99%

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

Iksanov¹,

Mallein²

2019

ALEA

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Lévy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form S := [0,∞) e −Xs− dZ s , where (X, Z) is a two-dimensional Lévy process, and Z is a drift-free Lévy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of S. This result and the previously known assertion due to Erickson and Maller [18] concerning the a.s. finiteness of S are then used to derive ultimate necessary and sufficient conditions for the L p -convergence for p > 1 and p = 1, respectively, of Biggins' martingales associated to branching Lévy processes. In particular, we provide final versions of results obtained recently by Bertoin and Mallein in [10].

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A renewal theorem and supremum of a perturbed random walk

Cited by 3 publications

References 18 publications

Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

Random walks in a moderately sparse random environment

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

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