Hitting Probabilities and Large Deviations

Let {Yn}n∈Z+ be a sequence of random variables in Rdand let$A \subset \mathbb{R}^d$. Then P{Yn∈ A for some n} is the hitting probability of the set A by the sequence {Yn}. We consider the asymptotic behavior, as m → ∞, of P{Yn∈ mA, some n} = P{hitting mA} whenever (1) the probability law of Yn/n sat...

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Veröffentlicht in:The Annals of probability 1996-10, Vol.24 (4), p.2065-2078
1. Verfasser: Collamore, Jeffrey F.
Format: Artikel
Sprache:eng
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Zusammenfassung:Let {Yn}n∈Z+ be a sequence of random variables in Rdand let$A \subset \mathbb{R}^d$. Then P{Yn∈ A for some n} is the hitting probability of the set A by the sequence {Yn}. We consider the asymptotic behavior, as m → ∞, of P{Yn∈ mA, some n} = P{hitting mA} whenever (1) the probability law of Yn/n satisfies the large deviation principle and (2) the central tendency of Yn/n is directed away from the given set A. For a particular function Ĩ, we show P{Yn∈ mA, some n} ≈ exp(-mĨ(A)).
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1041903218