Heavy traffic limit theorems in fluctuation theory
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Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.
1993 ◽
Vol 37
(3)
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pp. 553-557
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2004 ◽
Vol 41
(01)
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pp. 83-92
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1984 ◽
Vol 30
(2)
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pp. 169-173
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2006 ◽
Vol 15
(1)
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pp. 143-158
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1955 ◽
Vol 51
(1)
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pp. 92-95
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