Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks

Alsmeyer, Gerold; Iksanov, Alexander; Meiners, Matthias

doi:10.1007/s10959-012-0475-7

Cited by 16 publications

(47 citation statements)

References 46 publications

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“…provides a uniform approximation for (K * n (t)) t∈[0,1] which is tight enough to derive that (K * n (t)) t∈[0,1] , properly centered and normalized, converges weakly in the Skorohod space if the same is true for 1] . This new observation allows us to replace the existing methods based on Poissonization-de-Poissonization used in earlier works on the Bernoulli sieve and constitutes the first principal contribution of the present paper.…”

Section: Remark 23mentioning

confidence: 77%

“…The M 1 -topology is weaker than the J 1 -topology and M 1 -convergence of f n to f is equivalent to the convergence of the closed graph of f n to the closed graph of f . For instance, choosing f n (t) := 1 [1−1/n, 1+1/n) (t) + 2 · 1 [1+1/n, 2] (t) and f (t) := 2 · 1 [1,2] (t), the f n do converge to f in the M 1 -topology, but not in the J 1 -topology on D[0, 2]. Without going into details, we mention that the M 1 -topology is typically used in functional limit theorems in which the limit process has jumps unmatched in the convergent sequence of processes.…”

Section: J 1 -And M 1 -Topology: a Brief Reviewmentioning

confidence: 99%

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Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Alsmeyer

Iksanov

Marynych

2017

Stochastic Processes and their Applications

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The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D[0, 1] endowed with the J 1 -or M 1 -topology for the number K * n (t) of boxes containing at most [n t ] balls, t ∈ [0, 1], and the random distribution function K * n (t)/K * n (1), as n → ∞. The limit processes for K * n (t) are of the form (X(1) − X((1 − t)−)) t∈[0,1] , where X is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for K * n (t)/K * n (1) is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of K * n (1). First, for any Karlin occupancy scheme with deterministic probabilities (p k ) k≥1 , we obtain an approximation, uniformly in t ∈ [0, 1], of the number of boxes with at most [n t ] balls by a counting function defined in terms of (p k ) k≥1 . Second, we prove several FLTs for the number of visits to the interval [0, nt] by a perturbed random walk, as n → ∞. If the stick-breaking factor has a beta distribution with parameters θ > 0 and 1, the process (K * n (t)) t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [n t ] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel [5], Hansen [17], Donnelly, Kurtz and Tavaré [6], and Arratia and Tavaré [3].

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Section: Remark 23mentioning

confidence: 77%

Section: J 1 -And M 1 -Topology: a Brief Reviewmentioning

confidence: 99%

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Alsmeyer

Iksanov

Marynych

2017

Stochastic Processes and their Applications

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“…Let us note here in passing that this equality in law does no longer generally hold in Markovian environment. According to Theorem 2.1 in [10], Condition (3) is equivalent to the negative divergence of the PRW W n = log |Π n−1 | + log |B n |, n ≥ 0, which means that W n → −∞ a.s. (see also Section 12). This equivalence in turn is obtained by drawing on a fluctuation-theoretic result, stated as Theorem 3.1 in the next section, due to Spitzer [46] and Erickson [23].…”

Section: Random Difference Equations (Perpetuities) In Markovian Envimentioning

confidence: 99%

Fluctuation Theory for Markov Random Walks

Alsmeyer

Buckmann

2017

J Theor Probab

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Two fundamental theorems by Spitzer-Erickson and Kesten-Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk (S n ) n≥0 with iid increments X 1 , X 2 , . . . and the existence of moments of various related quantities like the first passage into (x, ∞) and the last exit time from (−∞, x] for arbitrary x ≥ 0 are studied in the Markov-modulated situation when the X n are governed by a positive recurrent Markov chain M = (M n ) n≥0 on a countable state space S, thus for a Markov random walk (M n , S n ) n≥0 . Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks (S τn(i) ) n≥0 , where τ 1 (i), τ 2 (i), . . . denote the successive return times of M to state i, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the afore-mentioned theorems are surprisingly more complicated and require the introduction of various excursion measures so as to characterize the existence of moments of different quantities.

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“…We investigate the two summands in the right-hand side separately. The second summand is a standard renewal shot noise process with response function h. Under condition (2) and assuming that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…represents the number of busy servers at time u ≥ 0. The process (Y (u)) u≥0 may also be interpreted as the difference between the number of visits to [0, t] of the standard random walk (S k ) k≥0 and the perturbed random walk (S k + η k+1 ) k≥1 , see [2], or as the number of active sources in a communication network, see [17,18]. An introduction to renewal theory for perturbed random walks can be found in [7].…”

Section: The Number Of Busy Servers In a G/g/∞ Queuementioning

confidence: 99%

A functional limit theorem for random processes with immigration in the case of heavy tails

Marynych¹,

Verovkin²

2017

Modern Stoch. Theory Appl.

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Let (X k , ξ k ) k∈N be a sequence of independent copies of a pair (X, ξ) where X is a random process with paths in the Skorokhod space D[0, ∞) and ξ is a positive random variable. The random process with immigration (Y (u)) u∈R is defined as the a.s. finite sum Y (u) = k≥0 X k+1 (u−ξ1 −· · ·−ξ k ) 1 {ξ 1 +···+ξ k ≤u} . We obtain a functional limit theorem for the process (Y (ut)) u≥0 , as t → ∞, when the law of ξ belongs to the domain of attraction of an α-stable law with α ∈ (0, 1), and the process X oscillates moderately around its mean E[X(t)]. In this situation the process (Y (ut)) u≥0 , when scaled appropriately, converges weakly in the Skorokhod space D(0, ∞) to a fractionally integrated inverse stable subordinator.Keywords Fractionally integrated inverse stable subordinators, random process with immigration, shot noise process 2010 MSC Primary 60F05, Secondary 60K05

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Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks

Cited by 16 publications

References 46 publications

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Fluctuation Theory for Markov Random Walks

A functional limit theorem for random processes with immigration in the case of heavy tails

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