scholarly journals On closure and factorization properties of subexponential and related distributions

1980 ◽  
Vol 29 (2) ◽  
pp. 243-256 ◽  
Author(s):  
Paul Embrechts ◽  
Charles M. Goldie
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Regularly Varying

AbstractFor a distribution function F on [0, ∞] we say F ∈ if {1 – F(2)(x)}/{1 – F(x)}→2 as x→∞, and F∈, if for some fixed γ > 0, and for each real , limx→∞ {1 – F(x + y)}/{1 – F(x)} ═ e– n. Sufficient conditions are given for the statement F ∈ F * G ∈ and when both F and G are in y it is proved that F*G∈pF + 1(1 – p) G ∈ for some (all) p ∈(0,1). The related classes ℒt are proved closed under convolutions, which implies the closure of the class of positive random variables with regularly varying tails under multiplication (of random variables). An example is given that shows to be a proper subclass of ℒ 0.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

On record values and the exponent of a distribution with regularly varying upper tail

1992 ◽  
Vol 29 (3) ◽  
pp. 575-586 ◽  
Author(s):  
Mohamed Berred
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Continuous Distribution ◽  
Random Variables ◽  
Record Values ◽  
Numerical Results ◽  
Continuous Distribution Function ◽  
Regularly Varying

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

On record values and the exponent of a distribution with regularly varying upper tail

1992 ◽  
Vol 29 (03) ◽  
pp. 575-586 ◽  
Author(s):  
Mohamed Berred
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Continuous Distribution ◽  
Random Variables ◽  
Record Values ◽  
Numerical Results ◽  
Continuous Distribution Function ◽  
Regularly Varying

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

scholarly journals Tail Behavior of Randomly Weighted Sums

2012 ◽  
Vol 44 (03) ◽  
pp. 794-814 ◽  
Author(s):  
Rajat Subhra Hazra ◽  
Krishanu Maulik
Keyword(s):  
Mellin Transform ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Positive Sequence ◽  
Weighted Sums ◽  
Tail Behavior ◽  
Moment Conditions ◽  
Randomly Weighted Sums ◽  
Regularly Varying

Let {X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t=1 ∞Θ t X t + (where X + = max{0, X}) and max1≤k<∞∑ t=1 k Θ t X t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1, X 2,… as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of {X t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑ t=1 ∞Θ t X t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X (∞) has a regularly varying tail then does X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t=1 ∞Θ t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

Functional normalizations for the branching process with infinite mean

1979 ◽  
Vol 16 (03) ◽  
pp. 513-525 ◽  
Author(s):  
Andrew D. Barbour ◽  
H.-J. Schuh
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Branching Process ◽  
Analogous Result ◽  
Sufficient Conditions ◽  
Probability Generating Function ◽  
Random Variables ◽  
Watson Process

It is well known that, in a Bienaymé-Galton–Watson process (Zn ) with 1 < m = EZ 1 < ∞ and EZ 1 log Z 1 <∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0< α < 1, it is possible to find functions U such that α n U (Zn ) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z 1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn ). The most stringent set of conditions reduces, when U (x) x, to the requirements EZ 1 = 1/α, EZ 1 log Z 1 <∞.

On the behavior of the failure rate and reversed failure rate in engineering systems

2020 ◽  
Vol 57 (3) ◽  
pp. 899-910
Author(s):  
Mahdi Tavangar
Keyword(s):  
Distribution Function ◽  
Failure Rate ◽  
Cumulative Distribution Function ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Coherent System ◽  
System Failure ◽  
Engineering Systems ◽  
Coherent Systems

AbstractIn this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.

scholarly journals Tail Behavior of Randomly Weighted Sums

2012 ◽  
Vol 44 (3) ◽  
pp. 794-814 ◽  
Author(s):  
Rajat Subhra Hazra ◽  
Krishanu Maulik
Keyword(s):  
Mellin Transform ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Positive Sequence ◽  
Weighted Sums ◽  
Tail Behavior ◽  
Moment Conditions ◽  
Randomly Weighted Sums ◽  
Regularly Varying

Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X(∞) = ∑t=1∞ΘtXt+ (where X+ = max{0, X}) and max1≤k<∞∑t=1kΘtXt, and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X1, X2,… as independent and identically distributed positive random variables. If X1 has a regularly varying tail of index -α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θt, t ≥ 1}, X(∞) = ∑t=1∞ΘtXt converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X(∞) has a regularly varying tail then does X1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑t=1∞Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

Functional normalizations for the branching process with infinite mean

1979 ◽  
Vol 16 (3) ◽  
pp. 513-525 ◽  
Author(s):  
Andrew D. Barbour ◽  
H.-J. Schuh
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Branching Process ◽  
Analogous Result ◽  
Sufficient Conditions ◽  
Probability Generating Function ◽  
Random Variables ◽  
Watson Process

It is well known that, in a Bienaymé-Galton–Watson process (Zn) with 1 < m = EZ1 < ∞ and EZ1 log Z1 <∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0< α < 1, it is possible to find functions U such that α n U (Zn) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn). The most stringent set of conditions reduces, when U (x) x, to the requirements EZ1 = 1/α, EZ1 log Z1 <∞.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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