Moments and limiting distribution of a portfolio of whole life annuity policies

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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The geometricity of the limiting distributions in queues and dams

Journal of Applied Probability ◽

10.1017/s0021900200117449 ◽

1993 ◽

Vol 30 (02) ◽

pp. 438-445

Author(s):

R. M. Phatarfod

Keyword(s):

Random Walks ◽

Limiting Distribution ◽

Duality Relation ◽

Limiting Distributions ◽

Initial Term

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

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QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations (Extended Abstract)

2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO) ◽

10.1137/1.9781611975505.9 ◽

2019 ◽

pp. 87-93

Author(s):

James Allen Fill ◽

Wei-Chun Hung

Keyword(s):

Large Deviations ◽

Limiting Distribution ◽

Tail Asymptotics

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On random divisions of a convex set

Journal of Applied Probability ◽

10.2307/3213129 ◽

1978 ◽

Vol 15 (3) ◽

pp. 645-649 ◽

Cited By ~ 3

Author(s):

Svante Janson

Keyword(s):

Elementary Proof ◽

Convex Set ◽

Limiting Distribution ◽

Independent Identically Distributed ◽

Normally Distributed

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

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Life Expectancy in Individuals With Type 2 Diabetes: Implications for Annuities

Medical Decision Making ◽

10.1177/0272989x09349960 ◽

2009 ◽

Vol 30 (3) ◽

pp. 409-414 ◽

Cited By ~ 5

Author(s):

Hermione C. Price ◽

Philip M. Clarke ◽

Alastair M. Gray ◽

Rury R. Holman

Keyword(s):

Type 2 Diabetes ◽

Life Expectancy ◽

Cholesterol Level ◽

Total Cholesterol Level ◽

Insurance Companies ◽

Age At Death ◽

Life Annuity ◽

Uk Government ◽

The Uk

Background. Insurance companies often offer people with diabetes ‘‘enhanced impaired life annuity’’ at preferential rates, in view of their reduced life expectancy. Objective. To assess the appropriateness of ‘‘enhanced impaired life annuity’’ rates for individuals with type 2 diabetes. Patients. There were 4026 subjects with established type 2 diabetes (but not known cardiovascular or other life-threatening diseases) enrolled into the UK Lipids in Diabetes Study. Measurements. Estimated individual life expectancy using the United Kingdom Prospective Diabetes Study (UKPDS) Outcomes Model. Results. Subjects were a mean (SD) age of 60.7 (8.6) years, had a blood pressure of 141/83 (17/10) mm Hg, total cholesterol level of 4.5 (0.75) mmol/L, HDL cholesterol level of 1.2 (0.29) mmol/L, with median (interquartile range [IQR]) known diabetes duration of 6 (3—11) years, and HbA1c of 8.0% (7.2—9.0). Sixty-five percent were male, 91% white, 4% Afro-Caribbean, 5% Indian-Asian, and 15% current smokers. The UKPDS Outcomes Model median (IQR) estimated age at death was 76.6 (73.8—79.5) years compared with 81.6 (79.4—83.2) years, estimated using the UK Government Actuary’s Department data for a general population of the same age and gender structure. The median (IQR) difference was 4.3 (2.8—6.1) years, a remaining life expectancy reduction of almost one quarter. The highest value annuity identified, which commences payments immediately for a 60-year-old man with insulin-treated type 2 diabetes investing 100,000, did not reflect this difference, offering 7.4K per year compared with 7.0K per year if not diabetic. Conclusions. The UK Government Actuary’s Department data overestimate likely age at death in individuals with type 2 diabetes, and at present, ‘‘enhanced impaired life annuity’’ rates do not provide equity for people with type 2 diabetes. Using a diabetes-specific model to estimate life expectancy could provide valuable information to the annuity industry and permit more equitable annuity rates for those with type 2 diabetes.

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The random division of faces in a planar graph

Advances in Applied Probability ◽

10.2307/1428040 ◽

1996 ◽

Vol 28 (2) ◽

pp. 331-331

Author(s):

Richard Cowan ◽

Simone Chen

Keyword(s):

Planar Graph ◽

Limiting Distribution ◽

Face Type ◽

Connected Planar Graph ◽

Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline

Journal of Applied Probability ◽

10.1017/s0021900200031466 ◽

1987 ◽

Vol 24 (03) ◽

pp. 758-767

Author(s):

D. Fakinos

Keyword(s):

Queueing System ◽

Limiting Distribution ◽

Single Server ◽

Queue Size ◽

Single Server Queue ◽

Preemptive Resume ◽

Server Queue ◽

Queue Discipline ◽

Service Times

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.

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Rate modulation in dams and ruin problems

Journal of Applied Probability ◽

10.2307/3215076 ◽

1996 ◽

Vol 33 (2) ◽

pp. 523-535 ◽

Cited By ~ 17

Author(s):

Søren Asmussen ◽

Offer Kella

Keyword(s):

State Space ◽

Release Rate ◽

Risk Process ◽

The State ◽

Limiting Distribution ◽

Process Conditions ◽

Explicit Formulas ◽

Rate Modulation ◽

Finite State ◽

Finite State Space

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

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