Asymptotic distribution of the maximum of n independent stochastic processes

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.

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Asymptotic properties of intensity estimators for Poisson shot-noise processes

Journal of Applied Probability ◽

10.1017/s002190020004242x ◽

1991 ◽

Vol 28 (03) ◽

pp. 568-583

Author(s):

Friedrich Liese ◽

Volker Schmidt

Keyword(s):

Asymptotic Normality ◽

Stochastic Processes ◽

Point Process ◽

Shot Noise ◽

Poisson Point Process ◽

Asymptotic Properties ◽

Weak Consistency ◽

Long Run ◽

Consistency And Asymptotic Normality ◽

Poisson Shot Noise

Stochastic processes {X(t)} of the form X(t) = Σ n f(t – Tn ) are considered, where {Tn } is a stationary Poisson point process with intensity λ and f: R → R is an unknown response function. Conditions are obtained for weak consistency and asymptotic normality of estimators of λ based on long-run observations of {X(t)}.

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Extreme values of independent stochastic processes

Journal of Applied Probability ◽

10.2307/3213346 ◽

1977 ◽

Vol 14 (4) ◽

pp. 732-739 ◽

Cited By ~ 113

Author(s):

Bruce M. Brown ◽

Sidney I. Resnick

Keyword(s):

Stochastic Processes ◽

Point Process ◽

Time Scales ◽

Poisson Point Process ◽

Point Processes ◽

Extreme Values ◽

Weak Limit ◽

One Dimensional ◽

Limit Process ◽

Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

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Extreme values of independent stochastic processes

Journal of Applied Probability ◽

10.1017/s0021900200105261 ◽

1977 ◽

Vol 14 (04) ◽

pp. 732-739 ◽

Cited By ~ 24

Author(s):

Bruce M. Brown ◽

Sidney I. Resnick

Keyword(s):

Stochastic Processes ◽

Point Process ◽

Time Scales ◽

Poisson Point Process ◽

Point Processes ◽

Extreme Values ◽

Weak Limit ◽

One Dimensional ◽

Limit Process ◽

Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

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Asymptotic properties of intensity estimators for Poisson shot-noise processes

Journal of Applied Probability ◽

10.2307/3214492 ◽

1991 ◽

Vol 28 (3) ◽

pp. 568-583 ◽

Cited By ~ 1

Author(s):

Friedrich Liese ◽

Volker Schmidt

Keyword(s):

Asymptotic Normality ◽

Stochastic Processes ◽

Point Process ◽

Shot Noise ◽

Poisson Point Process ◽

Asymptotic Properties ◽

Weak Consistency ◽

Long Run ◽

Consistency And Asymptotic Normality ◽

Poisson Shot Noise

Stochastic processes {X(t)} of the form X(t) = Σ n f(t – Tn) are considered, where {Tn} is a stationary Poisson point process with intensity λ and f: R → R is an unknown response function. Conditions are obtained for weak consistency and asymptotic normality of estimators of λ based on long-run observations of {X(t)}.

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Extremes of multitype branching random walks: heaviest tail wins

Advances in Applied Probability ◽

10.1017/apr.2019.20 ◽

2019 ◽

Vol 51 (2) ◽

pp. 514-540

Author(s):

Ayan Bhattacharya ◽

Krishanu Maulik ◽

Zbigniew Palmowski ◽

Parthanil Roy

Keyword(s):

Random Walk ◽

Random Walks ◽

Asymptotic Distribution ◽

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Weak Limit ◽

The Other ◽

Branching Random Walk ◽

Branching Random Walks

AbstractWe consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the nth generation.

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Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽

10.3390/math9050477 ◽

2021 ◽

Vol 9 (5) ◽

pp. 477

Author(s):

Katarzyna Górska ◽

Andrzej Horzela

Keyword(s):

Stochastic Processes ◽

Spectral Function ◽

Evolution Equations ◽

Memory Function ◽

Relaxation Phenomena ◽

Stieltjes Function ◽

Infinitely Divisible ◽

Spectral Functions ◽

Debye Relaxation ◽

Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

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