On the maximum partial sums of sequences of independent random variables

Chung, Kai Lai

doi:10.1090/s0002-9947-1948-0026274-0

Cited by 153 publications

(12 citation statements)

References 14 publications

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“…This generalizes Chung's law of iterated logarithm(Chung (1948)) and Theorem 3.3 of Monrad and RootzeÂ n (1995) to multiparameter cases.If X(t) is Brownian motion in R and we take 0Y 1, then (natural to believe that with the conditions of Theorem 3.1, the following inequality holds for any rectangle R x Hausdor measure of the level sets…”

supporting

confidence: 74%

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Xiao

1997

Probab Theory Relat Fields

144

172

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supporting

confidence: 74%

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Xiao

1997

Probab Theory Relat Fields

144

172

View full text Add to dashboard Cite

“…The first part is due to Shi [20] and requires a delicate analysis (which however is simpler than the calculations of Shi). The second statement is an immediate consequence of Chung's law of the iterated logarithm (see Chung [3]) and equations ( 6) and ( 7); we omit the details of its proof. On the other hand, we see by (11) that the series converges.…”

Section: ~00mentioning

confidence: 97%

Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process

Bertoin

Werner

1994

Lecture Notes in Mathematics

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L'accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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“…There are a number of other interesting a.s. properties of random walks one of which is the following due to Chung (1948): Suppose {ξ i } ∞ i=1 are i.i.d., mean-zero variance-one, and ξ 1 ∈ L 3 (P). Then s n = ξ 1 + · · · + ξ n satisfies lim inf n→∞ max 1≤j ≤n |s j | √ n/ ln ln n = π √ 8 a.s. (9.1) Chung (1948) contains the corresponding integral test.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Exceptional times and invariance for dynamical random walks

Khoshnevisan

Levin

Méndez-Hernández

2005

Probab. Theory Relat. Fields

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Consider a sequence {X i (0)} n i=1 of i.i.d. random variables.Associate to each X i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X i (t)} t≥0 (i = 1, 2, · · · ) whose invariant distribution is the law ν of X 1 (0). Benjamini et al. (2003) introduced the dynamical walk S n (t) = X 1 (t) + · · · + X n (t), and proved among other things that the LIL holds for n → S n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X i (0)'s are standard normal, the classical integral test is not dynamically stable.Presently, we study the set of times t when n → S n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ε-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate.We prove also that the infinite-dimensional process t → S n• (t)/ √ n converges weakly in D (D([0, 1])) to the Ornstein-Uhlenbeck process in C ([0, 1]). For this we assume only that the increments have mean zero and variance one.In addition, we extend a result of Benjamini et al. (2003) by proving that if the X i (0)'s are lattice, mean-zero variance-one, and possess (2 + ε) finite absolute moments for some ε > 0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest.

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On the maximum partial sums of sequences of independent random variables

Cited by 153 publications

References 14 publications

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process

Exceptional times and invariance for dynamical random walks

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