ISSN:
1572-9230
Keywords:
Random walks
;
first passage times
;
boundary crossing probabilities
;
sequential analysis
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let }S n} be a random walk, generated by i.i.d. increments X i which drifts weakly to ∞ in the sense that $$S_n \xrightarrow{P}\infty$$ as n→ ∞. Suppose k≥0, k≠1, and E|X 1|1k = ∞ if k〉1. Then we show that the probability that S. crosses the curve n↦an K before it crosses the curve n ↦ −an k tends to 1 as a → ∞. This intuitively plausible result is not true for k = 1, however, and for 1/2 〈k〈1, the converse results are not true in general, either. More general boundaries g(n) than g(n) = n k are also considered, and we also prove similar results for first passages out of regions like { (n, y): n≥1, |y| ≤ (a + n) k } as a→ ∞.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022621016708
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