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4395 results in Advances in Applied Probability

Page 210 of 220

  • Inverses of extremal processes

  • Sidney I. Resnick
  • Journal:
    Advances in Applied Probability / Volume 6 / Issue 2 / June 1974
    Published online by Cambridge University Press:
    01 July 2016, pp. 392-406
    Print publication:
    June 1974

  • The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

  • Renewal theory in two dimensions: Basic results

  • Jeffrey J. Hunter
  • Journal:
    Advances in Applied Probability / Volume 6 / Issue 2 / June 1974
    Published online by Cambridge University Press:
    01 July 2016, pp. 376-391
    Print publication:
    June 1974

  • In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

Page 210 of 220