Tail probabilities for non-standard risk and queueing processes with subexponential jumps

A well-known result on the distribution tail of the maximum of a random walk with heavy-tailed increments is extended to more general stochastic processes. Results are given in different settings, involving, for example, stationary increments and regeneration. Several examples and counterexamples illustrate that the conditions of the theorems can easily be verified in practice and are in part necessary. The examples include superimposed renewal processes, Markovian arrival processes, semi-Markov input and Cox processes with piecewise constant intensities.

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A unified approach to the study of tail probabilities of compound distributions

Journal of Applied Probability ◽

10.1239/jap/1032374755 ◽

1999 ◽

Vol 36 (4) ◽

pp. 1058-1073 ◽

Cited By ~ 19

Author(s):

Jun Cai ◽

José Garrido

Keyword(s):

Stochastic Ordering ◽

Renewal Processes ◽

Unified Approach ◽

Lower And Upper Bounds ◽

Tail Probabilities ◽

Compound Distributions ◽

Exponential Tails ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

Wald's Identity

We consider the tail probabilities of a class of compound distributions. First, the relations between reliability distribution classes and heavy-tailed distributions are discussed. These relations reveal that many previous results on estimating the tail probabilities are not applicable to heavy-tailed distributions.Then, a generalized Wald's identity and identities for compound geometric distributions are presented in terms of renewal processes. Using these identities, lower and upper bounds for the tail probabilities are derived in a unified way for the class of compound distributions, both under the conditions of NBU and NWU tails, which include exponential tails, as well as under the condition of heavy-tailed distributions.Finally, simplified bounds are derived by the technique of stochastic ordering. This method removes some unnecessary technical assumptions and corrects errors in the proof of some previous results.

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A unified approach to the study of tail probabilities of compound distributions

Journal of Applied Probability ◽

10.1017/s0021900200017861 ◽

1999 ◽

Vol 36 (04) ◽

pp. 1058-1073 ◽

Cited By ~ 1

Author(s):

Jun Cai ◽

José Garrido

Keyword(s):

Stochastic Ordering ◽

Renewal Processes ◽

Unified Approach ◽

Lower And Upper Bounds ◽

Tail Probabilities ◽

Compound Distributions ◽

Exponential Tails ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

Wald's Identity

We consider the tail probabilities of a class of compound distributions. First, the relations between reliability distribution classes and heavy-tailed distributions are discussed. These relations reveal that many previous results on estimating the tail probabilities are not applicable to heavy-tailed distributions. Then, a generalized Wald's identity and identities for compound geometric distributions are presented in terms of renewal processes. Using these identities, lower and upper bounds for the tail probabilities are derived in a unified way for the class of compound distributions, both under the conditions of NBU and NWU tails, which include exponential tails, as well as under the condition of heavy-tailed distributions. Finally, simplified bounds are derived by the technique of stochastic ordering. This method removes some unnecessary technical assumptions and corrects errors in the proof of some previous results.

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Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Queueing Systems ◽

10.1007/s11134-007-9051-8 ◽

2007 ◽

Vol 57 (2-3) ◽

pp. 115-127 ◽

Cited By ~ 21

Author(s):

S. Juneja

Keyword(s):

Relative Error ◽

Tail Probabilities ◽

Heavy Tailed Distributions ◽

Heavy Tailed

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On pair and tuple formation under independent Poisson or renewal arrival processes

Journal of Applied Probability ◽

10.1239/jap/1101840558 ◽

2004 ◽

Vol 41 (4) ◽

pp. 1138-1144 ◽

Cited By ~ 1

Author(s):

K. Borovkov

Keyword(s):

Weak Convergence ◽

Convergence Rate ◽

Formation Process ◽

Short Proof ◽

Probabilistic Approach ◽

Convergence Result ◽

Pair Formation ◽

Poisson Processes ◽

Renewal Processes ◽

Arrival Processes

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

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Maxima of Sums of Heavy-Tailed Random Variables

Astin Bulletin ◽

10.2143/ast.32.1.1013 ◽

2002 ◽

Vol 32 (1) ◽

pp. 43-55 ◽

Cited By ~ 37

Author(s):

K.W. Ng ◽

Q.H. Tang ◽

H. Yang

Keyword(s):

Asymptotic Properties ◽

Random Variables ◽

Independent Random Variables ◽

Partial Sums ◽

Tail Probabilities ◽

Large Classes ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

The Individual ◽

Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

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Statistical inference for point-process models of rainfall

Advances in Applied Probability ◽

10.1017/s0001867800022175 ◽

1984 ◽

Vol 16 (1) ◽

pp. 22-22 ◽

Cited By ~ 1

Author(s):

James A. Smtih ◽

Alan F. Karr

Keyword(s):

Parameter Estimation ◽

Statistical Inference ◽

Point Processes ◽

Process Models ◽

Renewal Processes ◽

Cox Processes ◽

The Family ◽

Point Process Models ◽

Image Position ◽

Rainfall Occurrences

In this paper we develop maximum likelihood procedures for parameter estimation and hypothesis testing for three classes of point processes that have been used to model rainfall occurrences; renewal processes, Neyman-Scott processes, and RCM processes (which are members of the family of Cox processes). The statistical inference procedures developed in this paper are based on the intensity process

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A versatile Markovian point process

Journal of Applied Probability ◽

10.2307/3213143 ◽

1979 ◽

Vol 16 (4) ◽

pp. 764-779 ◽

Cited By ~ 428

Author(s):

Marcel F. Neuts

Keyword(s):

Point Processes ◽

Probability Distributions ◽

Poisson Processes ◽

Renewal Processes ◽

Probability Models ◽

Phase Type ◽

Finite State ◽

Markov Point Processes ◽

Arrival Processes ◽

Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

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Risk Measures and Multivariate Extensions of Breiman's Theorem

Journal of Applied Probability ◽

10.1239/jap/1339878792 ◽

2012 ◽

Vol 49 (2) ◽

pp. 364-384 ◽

Cited By ~ 13

Author(s):

Anne-Laure Fougeres ◽

Cecile Mercadier

Keyword(s):

Ruin Probability ◽

Value At Risk ◽

Risk Measures ◽

Risk Measure ◽

Capital Requirements ◽

Regulatory Capital ◽

Tail Probabilities ◽

Solvency Capital ◽

Finite Time Horizon ◽

Standard Risk

The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.

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Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

Journal of Applied Probability ◽

10.1017/jpr.2020.85 ◽

2021 ◽

Vol 58 (1) ◽

pp. 217-237

Author(s):

Denis Denisov ◽

Elena Perfilev ◽

Vitali Wachtel

Keyword(s):

Random Walk ◽

Tail Asymptotics ◽

Tail Probabilities ◽

Tail Behaviour ◽

Negative Drift ◽

Heavy Tailed

AbstractWe study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

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