Skip to main content
×
Home
    • Aa
    • Aa

Uniform renewal theory with applications to expansions of random geometric sums

  • J. Blanchet (a1) and P. Glynn (a2)
Abstract

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Uniform renewal theory with applications to expansions of random geometric sums
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Uniform renewal theory with applications to expansions of random geometric sums
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Uniform renewal theory with applications to expansions of random geometric sums
      Available formats
      ×
Copyright
Corresponding author
Postal address: Statistics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA. Email address: blanchet@fas.harvard.edu
∗∗ Postal address: Management Science and Engineering, Stanford University, 380 Panama Way, Stanford, CA 94305, USA.
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J. Abate , G. L. Choudhury and W. Whitt (1995). Exponential approximations for tail probabilities in queues. I. Waiting times. Operat. Res. 43, 885901.

D. Aldous (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.

S. Asmussen (1984). Approximations for the probability of ruin within finite time. Scand. Actuarial J. 1984, 3157.

J. Blanchet and P. Glynn (2006). Corrected diffusion approximations for the maximum of light-tailed random walk. Ann. App. Prob. 16, 951953.

A. Borovkov and S. Foss (2000). Estimates for overshooting and arbitrary boundary by a random walk and their applications. Theory Prob. Appl. 44, 231253.

H. Carlsson (1983). Remainder term estimates of the renewal function. Ann. Prob. 11, 143157.

R. Durrett (2005). Probability: Theory and Examples. Duxbury, New York.

C. D. Fuh (2004). Uniform Markov renewal theory and ruin probabilities in Markov random walks. Ann. Appl. Prob. 14, 12021241.

T. H. Ganelius (1971). Tauberian Reminder Theorems (Lecture Notes Math. 232). Springer, Berlin.

M. L. Hogan (1986). Comment on: ‘Corrected diffusion approximations in certain random walk problems’. J. Appl. Prob. 23, 8996.

J. Keilson (1979). Rarity and Exponentiality. Springer, New York.

D. Siegmund (1979). Corrected diffusion approximations in certain random walk problems. Adv. Appl. Prob. 11, 701719.

D. Siegmund (1985). Sequential Analysis. Springer, New York.

C. Stone (1965). On moment generating functions and renewal theory. Ann. Math. Statist. 36, 12981301.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 23 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd March 2017. This data will be updated every 24 hours.