scholarly journals Tail probabilities for infinite series of regularly varying random vectors

Bernoulli ◽  
2008 ◽  
Vol 14 (3) ◽  
pp. 838-864 ◽  
Author(s):  
Henrik Hult ◽  
Gennady Samorodnitsky
Keyword(s):  
Infinite Series ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Regularly Varying

scholarly journals Stochastic Dominance and Weak Concentration for Sums of Independent Symmetric Random Vectors

2018 ◽  
Vol 2020 (23) ◽  
pp. 8997-9010
Author(s):  
Witold Bednorz ◽  
Tomasz Tkocz
Keyword(s):  
Stochastic Dominance ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Weak Concentration

Abstract Kwapień and Woyczyński asked in their monograph (1992) whether their notion of superstrong domination is inherited when taking sums of independent symmetric random vectors (one vector dominates another if, essentially, tail probabilities of any norm of the two vectors compare up to some scaling constants). We answer this question positively. As a by-product of our methods, we establish that a certain notion of weak concentration is also preserved by taking sums of independent symmetric random vectors.

scholarly journals Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

2015 ◽  
Vol 52 (1) ◽  
pp. 68-81 ◽  
Author(s):  
K. M. Kosiński ◽  
M. Mandjes
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Random Vectors ◽  
Deviation Principle ◽  
Logarithmic Asymptotics ◽  
Rate Functions ◽  
Regularly Varying ◽  
Positive Index

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

scholarly journals Moving Averages of Random Vectors with Regularly Varying Tails

2000 ◽  
Vol 21 (3) ◽  
pp. 297-328 ◽  
Author(s):  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Moving Averages ◽  
Random Vectors ◽  
Regularly Varying

scholarly journals On quadratic forms in multivariate generalized hyperbolic random vectors

Biometrika ◽  
2020 ◽  
Author(s):  
Simon A Broda ◽  
Juan Arismendi Zambrano
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Inversion Formula ◽  
Expected Shortfall ◽  
Distribution Functions ◽  
Least Squares Estimator ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Two Stage ◽  
Approximate Expressions

Summary This article presents exact and approximate expressions for tail probabilities and partial moments of quadratic forms in multivariate generalized hyperbolic random vectors. The derivations involve a generalization of the classic inversion formula for distribution functions (Gil-Pelaez, 1951). Two numerical applications are considered: the distribution of the two-stage least squares estimator and the expected shortfall of a quadratic portfolio.

On the distribution of multiple power series regularly varying at the boundary point

2019 ◽  
Vol 29 (6) ◽  
pp. 409-421 ◽  
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Boundary Point ◽  
Local Version ◽  
Random Vectors ◽  
Integral Limit ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Regularly Varying ◽  
Multiple Power Series

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

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