A functional limit theorem for general shot noise processes

Iksanov, Alexander; Rashytov, Bohdan

doi:10.1017/jpr.2019.95

Cited by 9 publications

(8 citation statements)

References 29 publications

(2 reference statements)

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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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“…There, the standing assumption is that X is an eventually nondecreasing deterministic function which is regularly varying at ∞ of nonnegative index. We stress that the techniques used in the present work and in [11] are very different.…”

Section: Introductionmentioning

confidence: 93%

“…In [11], another quite recent paper, functional limit theorems are proved for random processes with immigration at random times. There, the standing assumption is that X is an eventually nondecreasing deterministic function which is regularly varying at ∞ of nonnegative index.…”

Section: Introductionmentioning

confidence: 99%

Weak convergence of random processes with immigration at random times

Dong,

Iksanov

2019

Preprint

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By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. The so defined random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in [Iksanov et al. (2017). Bernoulli, 23, 1233-1278 and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

show abstract

Section: Introductionmentioning

confidence: 99%

“…There, the standing assumption is that X is an eventually nondecreasing deterministic function which is regularly varying at of nonnegative index. We stress that the techniques used in the present work and in [11] are very different.…”

Section: Introductionmentioning

confidence: 99%

Weak convergence of random processes with immigration at random times

Dong

Iksanov

2020

J. Appl. Probab.

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By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

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A functional limit theorem for general shot noise processes

Cited by 9 publications

References 29 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

Weak convergence of random processes with immigration at random times

Weak convergence of random processes with immigration at random times

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