Skip to main content
SpringerLink
Log in

Multi-parameter renewal processes and simulation of forest fires

  • Published:
Environmental and Ecological Statistics Aims and scope Submit manuscript

Abstract

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. Various properties of renewal processes are discussed. Renewal processes are proposed as a basis for modelling the spread of a forest fire under a prevailing wind.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abay A (1999). Renewal theorems for random walks in multidimensional time. Mathematica Slovaca 49: 371–380

    Google Scholar 

  • Carlsson H and Wainger S (1984). On the multidimensional renewal theorem. J Math Anal Appl 100: 316–322

    Article  Google Scholar 

  • Daley DJ and Vere-Jones D (1988). An introduction to the theory of point processes. Springer-Verlag, New York

    Google Scholar 

  • Galambos J, Indlekofer KH and Katai I (1987). A renewal theorem for random walks in multidimensional time. Trans Am Math Soc 300: 759–769

    Article  Google Scholar 

  • Hagwood C (1989). A renewal theorem in multidimensional time. Aust J Stat 31: 130–137

    Article  Google Scholar 

  • Hoglund T (1988). A multidimensional renewal theorem. Bulletin des Sciences Mathématiques 112: 111–138

    Google Scholar 

  • Hunter JJ (1974). Renewal theory in two dimensions: basic results. Adv Appl Prob 6: 376–391

    Article  Google Scholar 

  • Hunter JJ (1977). Renewal theory in two dimensions: bounds on the renewal function. Adv Appl Prob 9: 527–541

    Article  Google Scholar 

  • Ivanoff BG and Merzbach E (2000). Set-indexed martingales. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Ivanoff BG and Merzbach E (2006). What is a multi-parameter renewal process?. Stochastics 78: 411–441

    Google Scholar 

  • Ivanoff BG, Merzbach E and Plante M (2007). A compensator characterization of point processes on topological lattices. Electron J Probab 12: 47–74

    Google Scholar 

  • Klesov OI and Steinebach JG (1998). Asymptotic behavior of renewal processes defined by random walks with multidimensional time. Theory Prob Math Stat 56: 107–113

    Google Scholar 

  • Mazziotto G and Merzbach E (1988). Point processes indexed by directed sets. Stochastic Processes and their Applications 30: 105–119

    Article  Google Scholar 

  • Nagaev AV (1979). Renewal theorems in R d. Theory Prob Appl 24: 572–581

    Article  Google Scholar 

  • Spitzer F (1986) A multidimensional renewal theorem. Probability,statistical mechanics, and number theory, advances in mathematics,supplementary studies 9:147–155, Academic Press, Orlando, FL

  • Turner M and Romme WH (1994). Landscape dynamics in crown fire ecosystems. Landsc Ecol 9: 59–77

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON, Canada, K1N 6N5

    B. Gail Ivanoff

Authors
  1. B. Gail Ivanoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Gail Ivanoff.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivanoff, B.G. Multi-parameter renewal processes and simulation of forest fires. Environ Ecol Stat 16, 153–171 (2009). https://doi.org/10.1007/s10651-007-0082-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10651-007-0082-4

Keywords

Search

Navigation