Precise asymptotics in the law of the iterated logarithm

Gut, Allan; Spătaru, Aurel

doi:10.1214/aop/1019160511

Cited by 113 publications

(19 citation statements)

References 13 publications

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“…Proof. The first result is a one-sided special case of [17], Theorem 1. The remaining part of the proof is similar to the corresponding parts of the previous one.…”

mentioning

confidence: 92%

“…By the same approach it is possible to prove the following result, the first conclusion of which is a one-sided version of [17], Theorem 2. An analogous convergence rate result could, of course, also have been included.…”

mentioning

confidence: 93%

“…The remaining part of the proof is similar to the corresponding parts of the previous one. 2 Remark 7.1 In a recent preprint Spȃtaru [27], Remark 8, mentions that it can be shown that Theorem 1 of [17], in fact, holds under condition (7.6) with δ = 0. Oleg Klesov [23] has, independently, made the same observation.…”

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confidence: 99%

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Convergence rates and precise asymptotics for renewal counting processes and some first passage times

Gut¹,

Steinebach²

2004

Asymptotic Methods in Stochastics

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Let X 1 , X 2 , . . . be i.i.d. random variables with mean 0 and common distribution function F , and set S n = X 1 +X 2 +...+X n , n ≥ 1. In recent years precise asymptotics as ε 0 have been proved for sums like ∞ n=1 n r/p−2 P (|S n | ≥ εn 1/p ). Our main results are analogs for renewal counting processes and first passage time processes of random walks. In the latter setup we consider the case E X > 0 as well as the case E X = 0.

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“…Proof. The first result is a one-sided special case of [17], Theorem 1. The remaining part of the proof is similar to the corresponding parts of the previous one.…”

mentioning

confidence: 92%

mentioning

confidence: 93%

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Convergence rates and precise asymptotics for renewal counting processes and some first passage times

Gut¹,

Steinebach²

2004

Asymptotic Methods in Stochastics

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“…For a sequence of independent and identically distributed nondegenerate random variables fX; X n ; n $ 1} with EX ¼ 0 and EX 2 ¼ s 2 , this question has been studied thoroughly by many authors. For example, Gut and Spâtaru [12] established that, for S n :¼ P n i¼1 X i ,…”

Section: Resultsmentioning

confidence: 99%

Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

Jiang

Xie

2015

Stochastics

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We study the asymptotic behaviours for the trajectory fitting estimator in the OrnsteinUhlenbeck process with linear drift. Deviation inequality, moderate deviations, BerryEsseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry-Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals [P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966-988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448-505].

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“…The first result in this direction due to Heyde [3], who proved that lim ↘0 2 ∑ ∞ =1 (| | ≥ ) = E 2 under the assumptions that E = 0 and E 2 < ∞. Some analogous results in more general case can be found in Gut and Spǎtaru [4,5] and Gut and Steinebach [6]. Moreover, Chow [7] studied the convergence properties of the series…”

Section: Lemma 2 Assume the Law Of The Entries {Re( ) Im( )mentioning

confidence: 96%

Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

Xie

2013

Abstract and Applied Analysis

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This paper considers the precise asymptotics of the spectral statistics of random matrices. Following the ideas of Gut and Spătaru (2000) and Liu and Lin (2006) on the precise asymptotics of i.i.d. random variables in the context of the complete convergence and the second-order moment convergence, respectively, we will establish the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of Wigner matrices or sample covariance matrices.

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Precise asymptotics in the law of the iterated logarithm

Cited by 113 publications

References 13 publications

Convergence rates and precise asymptotics for renewal counting processes and some first passage times

Convergence rates and precise asymptotics for renewal counting processes and some first passage times

Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

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