On the Distribution of the First Passage Time of a Random Walk to an Arbitrary Remote Boundary

Боровков, А. А.

doi:10.1137/s0040585x97t988125

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The Proof Of the Main Results2

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Cited by 3 publications

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“…Chapter 9 in [4]). They also proved instrumental for studying the distribution of the first passage time of a curvilinear boundary by a random walk [5], establishing integro-local theorems for compound renewal processes (see Chapter 10 in [4]) and in a number of other problems.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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A refined version of the integro-local Stone theorem

Боровков

Borovkov

2017

Statistics & Probability Letters

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Let X, X 1 , X 2 , . . . be a sequence of non-lattice i.i.d. random variables with E X = 0, E X = 1, and let S n := X 1 + · · · + X n , n 1. We refine Stone's integro-local theorem by deriving the first term in the asymptotic expansion for the probability P S n ∈ [x, x + ∆) as n → ∞ and establishing uniform bounds for the remainder term, under the assumption that the distribution of X satisfies Cramér's strong non-lattice condition and E |X| r < ∞ for some r 3.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…Now returning to (5) and using representation (17) for I 1 and the bounds (18), (19) for I 2 and I 3 , we obtain that, for ∆ ∈ [q n , ∆ 1 ] with an arbitrary fixed ∆ 1 0, one has…”

Section: The Proof Of the Main Resultsmentioning

confidence: 99%