Weak convergence, point processes and regular variation

1980 ◽  
Vol 12 (2) ◽  
pp. 296-296
Author(s):  
Sidney Resnick
Keyword(s):  
Weak Convergence ◽  
Regular Variation ◽  
Point Processes ◽  
Convergence Point

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (1) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

A note on disjoint-occurrence inequalities for marked Poisson point processes

1996 ◽  
Vol 33 (2) ◽  
pp. 420-426 ◽  
Author(s):  
J. van den Berg
Keyword(s):  
Weak Convergence ◽  
Point Processes ◽  
Poisson Point Processes ◽  
Bk Inequality

For (marked) Poisson point processes we give, for increasing events, a new proof of the analog of the BK inequality. In contrast to other proofs, which use weak-convergence arguments, our proof is ‘direct' and requires no extra topological conditions on the events. Apart from some well-known properties of Poisson point processes, the proof is self-contained.

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.

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.

Extreme Values, Regular Variation, and Point Processes.

1989 ◽  
Vol 84 (407) ◽  
pp. 845
Author(s):  
Alan Kimber ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Point Processes ◽  
Extreme Values

On Stably Weak Convergence of Semi-Martingales and of Point Processes

1984 ◽  
Vol 28 (2) ◽  
pp. 337-350 ◽  
Author(s):  
B. Grigelionis ◽  
R. Mikulyavichus
Keyword(s):  
Weak Convergence ◽  
Point Processes

Fluctuation theory in continuous time

1975 ◽  
Vol 7 (04) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

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