Summary
LetZ k ≡ (X k ,Y k ), k = 1,2,..., whereY k ∈R r−1, be independent with common distribution,E{|X 1|} < ∞,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 xn ,S yn ) =Z 1 + ··· +Z n . If\(U(A)\mathop = \limits^{df} \sum\limits_m {P\left\{ {S_m \in A} \right\}}\),
whereλ(t) + (μ −1 t)−1/α, andX 1 is nonarithmetic,W t converges to the product of Lebesgue measure andB. IfN (t) is the epoch of first entrance into {x≧t} by theS n, the distribution ofS x N(t)-t,λ(t)Sy N(t) converges to the product ofR andB, whereR is the well-known limiting distribution ofS x N(t)—t. Similar results are obtained for arithmeticX k.
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References
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Stam, A.J. Two theorems inr-dimensional renewal theory. Z. Wahrscheinlichkeitstheorie verw Gebiete 10, 81–86 (1968). https://doi.org/10.1007/BF00572924
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DOI: https://doi.org/10.1007/BF00572924