On large deviations

Statulevičius, V.

doi:10.1007/bf00537136

Cited by 122 publications

(37 citation statements)

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“…The subsequent Theorem 2 is derived from (1.4) combined with a well-known lemma on large deviations for a single random variable due to Statulevičius [15], see also [13], Lemma 2.3. In this way we obtain large deviations relations (in the sense of H. Cramér) for the total length of the BM contained in [0, T ] as well as an optimal uniform bound of the distance between the distribution function F T (x) = P(| ∩ [0, T ]| − E| ∩ [0, T ]| ≤ xσ T ) and the standard normal distribution function (x) = …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line

Heinrich

2007

Acta Appl Math

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We consider a not necessarily stationary one-dimensional Boolean model = i≥1 ( i + X i ) defined by a Poisson process = i≥1 δ X i with bounded intensity function λ(t) ≤ λ 0 and a sequence of independent copies 1 , 2 , . . . of a random compact subset 0 of the real line R 1 whose diameter 0 possesses a finite exponential moment E exp{a 0 }. We first study the higher-order covariance functions E ξ(t 1 )ξ(t 2 ) · · · ξ(t k ) of the {0, 1}-valued stochastic process ξ(t) = 1 c (t) , t ∈ R 1 , and derive exponential estimates of them as well as of the mixed cumulants Cum k (ξ(t 1 ), ξ(t 2 ), . . . , ξ(t k )). From this, we derive Cramér-type large deviations relations and a Berry-Esseen bound for the distribution of empirical total length meas( ∩ [0, T ]) of within [0, T ] as T grows large. Second, we prove that the family of events {ξ(t) = 1} = {t / ∈ }, t ∈ R 1 , satisfies an almost-Markovtype mixing condition with an exponentially decaying mixing rate. In case of a stationary Boolean model, i.e. λ(t) ≡ λ 0 , these properties enable us to show the existence and analyticity of the thermodynamic limitKeywords Poisson grain model on the line · Total length of clumps · Mixed cumulants · Higher-order covariances · Thermodynamic limit · Cramér-type large deviations relations · Berry-Esseen bound Mathematics Subject Classification (2000)Primary 60D05, 60G55 · Secondary 60G60, 62F05

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Section: Introduction and Main Resultsmentioning

confidence: 99%

An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line

Heinrich

2007

Acta Appl Math

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“…One of the fundamentally new steps in the theory of large deviations was taken by Statulevi6ius [24], who proposed to assume conditions on the behavior of cumulants. For one variable ~ (E/I = 0, D~ = 1), the Statulevi~ius condition (S) can be written in the following way:…”

Section: Resultsmentioning

confidence: 99%

Large deviations for integer centered poisson approximation

Čekanavičius¹,

Vaitkus²

1999

Lith Math J

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“…Obviously, (3) is a lattice analogue of the Statulevieius (S) condition for cumulants [36]. Condition (S) and its generalizations (see [34]) appeared to be very useful for the normal approximation, because it reduced the problem of large deviations to the problem of estimating such "nice" quantities as cumulants (a comprehensive review of the results under condition (S) can be found in the book by Saulis and Statulevieius [35]).…”

Section: Integral and Local Estimatesmentioning

confidence: 98%