The tail behaviour of a random sum of subexponential random variables and vectors

Extremes ◽  
2007 ◽  
Vol 10 (1-2) ◽  
pp. 21-39 ◽  
Author(s):  
D. J. Daley ◽  
Edward Omey ◽  
Rein Vesilo
Keyword(s):  
Random Variables ◽  
Random Sum ◽  
Tail Behaviour

Large deviations of heavy-tailed random sums with applications in insurance and finance

1997 ◽  
Vol 34 (2) ◽  
pp. 293-308 ◽  
Author(s):  
C. Klüppelberg ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Large Deviation ◽  
Random Variables ◽  
Compound Poisson Process ◽  
Random Sum ◽  
Random Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Heavy Tailed ◽  
The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Asymptotic tail behaviour of phase-type scale mixture distributions

2017 ◽  
Vol 12 (2) ◽  
pp. 412-432 ◽  
Author(s):  
Leonardo Rojas-Nandayapa ◽  
Wangyue Xie
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Mixture Distributions ◽  
Scale Mixture ◽  
Domains Of Attraction ◽  
Heavy Tailed Distributions ◽  
Tail Behaviour ◽  
Phase Type ◽  
Heavy Tailed

AbstractWe consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable Y and a non-negative but otherwise arbitrary random variable S called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable S has discrete support – such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples.

Perpetuities with thin tails

1996 ◽  
Vol 28 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Charles M. Goldie ◽  
Rudolf Grübel
Keyword(s):  
Difference Equations ◽  
Random Variables ◽  
Random Variable ◽  
Probabilistic Algorithms ◽  
Stochastic Difference Equations ◽  
Tail Behaviour ◽  
Independent Identically Distributed

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

scholarly journals Approximating the Distribution of a Dynamic Risk Portfolio

1984 ◽  
Vol 14 (2) ◽  
pp. 135-148 ◽  
Author(s):  
William S. Jewell
Keyword(s):  
Recursive Algorithm ◽  
Random Variables ◽  
Approximation Procedure ◽  
Random Sum ◽  
Dynamic Risk ◽  
Sum Of Random Variables

AbstractIn a previous paper, Jewell and Sundt showed how to approximate a distribution of total losses from a large, fixed, heterogeneous portfolio, using a recursive algorithm developed by Panjer for the distribution of a random sum of random variables (a single casualty contract). This paper extends the approximation procedure to large, dynamic heterogeneous portfolios, in order to model either a portfolio of correlated casualty contracts, or a future portfolio, whose composition is not known with certainty.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (02) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Large deviations of heavy-tailed random sums with applications in insurance and finance

1997 ◽  
Vol 34 (02) ◽  
pp. 293-308 ◽  
Author(s):  
C. Klüppelberg ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Large Deviation ◽  
Random Variables ◽  
Compound Poisson Process ◽  
Random Sum ◽  
Random Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Heavy Tailed ◽  
The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

An expansion for the distribution function of a random sum

1973 ◽  
Vol 10 (4) ◽  
pp. 869-874 ◽  
Author(s):  
L. M. Marsh
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Special Form ◽  
Edgeworth Expansion ◽  
Random Number ◽  
Probability Generating Function ◽  
Random Variables ◽  
Fixed Number ◽  
Random Sum

The Edgeworth expansion gives an indication of the rate of convergence of the distribution function of the sum of a fixed number of random variables to the normal distribution. A similar expansion is given here for the distribution function of the sum of a random number N of random variables, when the probability generating function of N takes a special form.

Ageing properties of certain dependent geometric sums

1992 ◽  
Vol 29 (3) ◽  
pp. 655-666 ◽  
Author(s):  
Antal Kováts ◽  
Tamás F. Móri
Keyword(s):  
Failure Time ◽  
Random Variables ◽  
Working Time ◽  
System Failure ◽  
Random Sum ◽  
Unit System ◽  
Repair Facility ◽  
Geometric Sums ◽  
First Time ◽  
Operating Unit

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

scholarly journals On the random functional central limit theorems with almost sure convergence for subsequences

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Zdzisław Rychlik ◽  
Konrad S. Szuster
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Random Sum ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractIn this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (2) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

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