scholarly journals Convergence to Stable Laws in the Space D

2015 ◽  
Vol 52 (01) ◽  
pp. 1-17 ◽  
Author(s):  
François Roueff ◽  
Philippe Soulier
Keyword(s):  
Point Process ◽  
Regular Variation ◽  
Metric Spaces ◽  
Empirical Process ◽  
Stable Distributions ◽  
Stable Laws ◽  
Random Elements ◽  
Process Convergence ◽  
Reward Process ◽  
Renewal Reward Process

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the spaceDof càdlàg functions endowed with Skorokhod'sJ1topology, to stable distributions inD. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

scholarly journals Convergence to Stable Laws in the SpaceD

2015 ◽  
Vol 52 (1) ◽  
pp. 1-17 ◽  
Author(s):  
François Roueff ◽  
Philippe Soulier
Keyword(s):  
Point Process ◽  
Regular Variation ◽  
Metric Spaces ◽  
Empirical Process ◽  
Stable Distributions ◽  
Stable Laws ◽  
Random Elements ◽  
Process Convergence ◽  
Reward Process ◽  
Renewal Reward Process

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the spaceDof càdlàg functions endowed with Skorokhod'sJ1topology, to stable distributions inD. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

scholarly journals Stable laws and Beurling kernels

2016 ◽  
Vol 48 (A) ◽  
pp. 239-248 ◽  
Author(s):  
Adam J. Ostaszewski
Keyword(s):  
Regular Variation ◽  
Close Relation ◽  
Direct Analysis ◽  
Stable Distributions ◽  
Stable Laws

AbstractWe identify a close relation between stable distributions and the limiting homomorphisms central to the theory of regular variation. In so doing some simplifications are achieved in the direct analysis of these laws in Pitman and Pitman (2016); stable distributions are themselves linked to homomorphy.

scholarly journals The density of a passage time for a renewal-reward process perturbed by a diffusion

2013 ◽  
Vol 26 (1) ◽  
pp. 108-112 ◽  
Author(s):  
Christophette Blanchet-Scalliet ◽  
Diana Dorobantu ◽  
Didier Rullière
Keyword(s):  
Passage Time ◽  
Reward Process ◽  
Renewal Reward Process

Sample quantiles of additive renewal reward processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1018-1032 ◽  
Author(s):  
Angelos Dassios
Keyword(s):  
Brownian Motion ◽  
Random Processes ◽  
Independent Increments ◽  
Look Back ◽  
Renewal Reward Processes ◽  
Reward Process ◽  
Renewal Reward Process ◽  
Sample Quantiles

The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.

scholarly journals A formula for hidden regular variation behavior for symmetric stable distributions

Extremes ◽  
2020 ◽  
Vol 23 (4) ◽  
pp. 667-691
Author(s):  
Malin Palö Forsström ◽  
Jeffrey E. Steif
Keyword(s):  
Power Law ◽  
Spectral Measure ◽  
Regular Variation ◽  
Stable Distributions ◽  
Decay Rates ◽  
Random Vectors ◽  
Power Law Decay ◽  
Exact Power ◽  
Hidden Regular Variation ◽  
Stable Random Vectors

Abstract We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (3) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

Policy Improvement and the Newton–Raphson Algorithm for Renewal Reward Processes

1989 ◽  
Vol 3 (3) ◽  
pp. 393-396 ◽  
Author(s):  
J. M. McNamara
Keyword(s):  
Continuous Time ◽  
Successive Approximations ◽  
Average Reward ◽  
Policy Improvement ◽  
Unique Root ◽  
Renewal Reward Processes ◽  
Reward Process ◽  
Newton Raphson ◽  
Renewal Reward Process ◽  
Raphson Algorithm

We consider a renewal reward process in continuous time. The supremum average reward, γ* for this process can be characterised as the unique root of a certain function. We show how one can apply the Newton–Raphson algorithm to obtain successive approximations to γ*, and show that the successive approximations so obtained are the same as those obtained by using the policy improvement technique.

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