Weak convergence of random processes with immigration at random times

Dong, Chuanfei; Iksanov, Alexander

doi:10.1017/jpr.2019.88

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…are independent copies of (N * j−1 (t)) t≥0 which are also independent of T * . Note that, for j ≥ 2, (N * j (t)) t≥0 is a particular instance of a random process with immigration at random times (the term was introduced in [9], see also [21]).…”

Section: Limit Theorems For a Special Branching Random Walkmentioning

confidence: 99%

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

Buraczewski

Dovgay²,

Iksanov³

2020

Electron. J. Probab.

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Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a 'box'. Given the weights 'balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the jth generation, independently of the others, hits a daughter box in the (j + 1)th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment.Assuming that the stick-breaking factor has a uniform distribution on [0, 1] and that the number of balls is n we investigate occupancy of intermediate generations, that is, those with indices jnu for u > 0, where jn diverges to infinity at a sublogarithmic rate as n becomes large. Denote by Kn(j) the number of occupied (ever hit) boxes in the jth generation. It is shown that the finite-dimensional distributions of the process (Kn( jnu ))u>0, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

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Section: Limit Theorems For a Special Branching Random Walkmentioning

confidence: 99%

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

Buraczewski

Dovgay²,

Iksanov³

2020

Electron. J. Probab.

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“…Besides being of independent interest, our findings pave the way towards controlling the asymptotic behavior of the second summand in (1). These, taken together with the results from [11], should eventually lead to understanding of the asymptotics of processes Y .…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…The summands should be treated separately, for each of these requires a specific approach. Weak convergence of the first summand in (1) is investigated in [11].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A functional limit theorem for general shot noise processes

Iksanov

Rashytov

2020

J. Appl. Probab.

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By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

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“…Note that, for j ≥ 2, N j is a particular instance of a random process with immigration at random times (the term was introduced in [8]; see also [15]).…”

Section: Introductionmentioning

confidence: 99%

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

et al. 2022

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An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

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Weak convergence of random processes with immigration at random times

Cited by 7 publications

References 11 publications

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

A functional limit theorem for general shot noise processes

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

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