Tail probabilities of sums of random vectors in banach spaces, and related mixed norms

1980 ◽  
pp. 455-469 ◽  
Author(s):  
Wojbor A. Woyczyński
Keyword(s):  
Banach Spaces ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Mixed Norms

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 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 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.

APPROXIMATION OF THE TAIL PROBABILITIES FOR BIDIMENSIONAL RANDOMLY WEIGHTED SUMS WITH DEPENDENT COMPONENTS

2018 ◽  
Vol 34 (1) ◽  
pp. 112-130
Author(s):  
Xinmei Shen ◽  
Mingyue Ge ◽  
Ke-Ang Fu
Keyword(s):  
Discrete Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Moment Conditions ◽  
Randomly Weighted Sums ◽  
Asymptotic Formulae ◽  
Extended Regular Variation

AbstractLet $\left\{ {{\bi X}_k = {(X_{1,k},X_{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of independent and identically distributed random vectors whose components are allowed to be generally dependent with marginal distributions being from the class of extended regular variation, and let $\left\{ {{\brTheta} _k = {(\Theta _{1,k},\Theta _{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of nonnegative random vectors that is independent of $\left\{ {{\bi X}_k, k \ge 1} \right\}$. Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums $\left( {\sum\nolimits_{k = 1}^n {\Theta _{1,k}} X_{1,k},\sum\nolimits_{k = 1}^n {\Theta _{2,k}} X_{2,k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{1,k}} X_{1,k},{{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{2,k}} X_{2,k})^{\rm \top }$ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on $\left\{ {{\brTheta} _k, k \ge 1} \right\}$. Direct applications of the results to risk analysis are proposed, with two types of ruin probability for a discrete-time bidimensional risk model being evaluated.

Weak Convergence of Empirical Processes of Dependent Random Vectors Under Logarithmic Mixing Rates

1992 ◽  
Vol 42 (1-2) ◽  
pp. 111-118
Author(s):  
Chandrakant M. Deo
Keyword(s):  
Weak Convergence ◽  
Gaussian Processes ◽  
Empirical Processes ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Mixing Rates

It is proved that empirical processes of ø-mixing or Ψ-mlxing random vectors conv11rge to appropriate Gaussian processes under logarithmic mixing rates. A Kiefer-type bound on the tail-probabilities of empirical processes is also obtained under the same mixing rates.

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