Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
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.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (04) ◽  
pp. 732-739 ◽  
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.

scholarly journals Extremes of multitype branching random walks: heaviest tail wins

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.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (03) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (3) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (3) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

The distribution of the convex hull of a Gaussian sample

1980 ◽  
Vol 17 (03) ◽  
pp. 686-695 ◽  
Author(s):  
William F. Eddy
Keyword(s):  
Weak Convergence ◽  
Convex Hull ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Sets ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Limit Process ◽  
Gaussian Sample

The distribution of the convex hull of a random sample ofd-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

Asymptotic properties of intensity estimators for Poisson shot-noise processes

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)}.

Asymptotic distribution of the maximum of n independent stochastic processes

1993 ◽  
Vol 30 (1) ◽  
pp. 66-81 ◽  
Author(s):  
A. A. Balkema ◽  
L. De Haan ◽  
R. L. Karandikar
Keyword(s):  
Stochastic Processes ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Poisson Point Process ◽  
Spectral Functions ◽  
Distribution Of The Maximum

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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