scholarly journals Nonexponential asymptotics for the solutions of renewal equations, with applications

2006 ◽  
Vol 43 (03) ◽  
pp. 815-824 ◽  
Author(s):  
Chuancun Yin ◽  
Junsheng Zhao
Keyword(s):  
Distribution Function ◽  
Birth Rate ◽  
Geometric Distribution ◽  
Renewal Processes ◽  
Renewal Function ◽  
Renewal Equations ◽  
Put Option ◽  
Special Cases ◽  
Compound Geometric Distribution ◽  
Defective Renewal Equations

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

scholarly journals Nonexponential asymptotics for the solutions of renewal equations, with applications

2006 ◽  
Vol 43 (3) ◽  
pp. 815-824 ◽  
Author(s):  
Chuancun Yin ◽  
Junsheng Zhao
Keyword(s):  
Distribution Function ◽  
Birth Rate ◽  
Geometric Distribution ◽  
Renewal Processes ◽  
Renewal Function ◽  
Renewal Equations ◽  
Put Option ◽  
Special Cases ◽  
Compound Geometric Distribution ◽  
Defective Renewal Equations

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (03) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (3) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Filtering of Markov renewal queues, IV: Flow processes in feedback queues

1985 ◽  
Vol 17 (2) ◽  
pp. 386-407 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Flow Process ◽  
Special Cases ◽  
Flow Processes ◽  
Transition Points ◽  
The Times

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

scholarly journals On a general class of renewal risk process: analysis of the Gerber-Shiu function

2005 ◽  
Vol 37 (03) ◽  
pp. 836-856 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Process Analysis ◽  
Risk Process ◽  
Expected Discounted Penalty Function ◽  
Renewal Equations ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
Renewal Risk ◽  
Defective Renewal Equations ◽  
Expected Discounted Penalty

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

scholarly journals Stein's Method for Compound Geometric Approximation

2010 ◽  
Vol 47 (1) ◽  
pp. 146-156 ◽  
Author(s):  
Fraser Daly
Keyword(s):  
Generating Functions ◽  
Analytical Approach ◽  
Geometric Distribution ◽  
Stein’S Method ◽  
Hitting Times ◽  
Stein's Method ◽  
Geometric Approximation ◽  
Compound Geometric Distribution ◽  
Probabilistic Approximation ◽  
Sequence Patterns

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

A general shock model for a reliability system

2000 ◽  
Vol 37 (04) ◽  
pp. 925-935 ◽  
Author(s):  
Georgios Skoulakis
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Renewal Process ◽  
General System ◽  
System Structure ◽  
Random Numbers ◽  
System Failure ◽  
Shock Model ◽  
Stieltjes Transform ◽  
Special Cases

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

Characterizations of the exponential distribution by order statistics

1974 ◽  
Vol 11 (03) ◽  
pp. 605-608 ◽  
Author(s):  
J. S. Huang
Keyword(s):  
Distribution Function ◽  
Order Statistics ◽  
Exponential Distribution ◽  
Special Cases ◽  
Characterization Theorems

Let X 1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX 1,n is identically distributed as X 1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

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