Two theorems inr-dimensional renewal theory

  • A. J. Stam
Article

Summary

LetZ k ≡ (X k ,Y k ), k = 1,2,..., whereY k Rr−1, be independent with common distribution,E{|X1|} < ∞,E {X I ) =μ > 0 and theY-distribution belonging to the domain of uncentered normal attraction of a stable lawB with exponent α ∃ (0, 2]. LetS n ≡ (S n x ,S n y ) =Z1 + ··· +Z n . If\(U(A)\mathop = \limits^{df} \sum\limits_m {P\left\{ {S_m \in A} \right\}}\),
$$\smallint gdW_t \mathop = \limits^{df} \mu \smallint g\left( {x - t,\lambda \left( t \right)y} \right)dU\left( {x,y} \right),$$
whereλ(t) + (μ−1t)−1/α, andX1 is nonarithmetic,W t converges to the product of Lebesgue measure andB. IfN (t) is the epoch of first entrance into {x≧t} by theSn, the distribution ofSxN(t)-t,λ(t)SyN(t) converges to the product ofR andB, whereR is the well-known limiting distribution ofSxN(t)t. Similar results are obtained for arithmeticXk.

Keywords

Stochastic Process Probability Theory Lebesgue Measure Mathematical Biology Common Distribution 

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References

  1. 1.
    Bickel, P. J., andJ. A. Yahav: Renewal theory in the plane. Ann. math. Statistics36, 946–955 (1965).Google Scholar
  2. 2.
    Feller, W.: An introduction to probability theory and its applications, Vol. II. London-New York: Wiley 1966.Google Scholar
  3. 3.
    Rényi, A.: Wahrscheinlichkeitsrechnung. Berlin: VEB Deutscher Verlag der Wissenschaften 1962.Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • A. J. Stam
    • 1
  1. 1.Mathematisch Instituut der RijksuniversiteitGroningenNetherlands

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