Small ball probabilities for stable convolutions

Аурзада, Франк; Simon, Thomas

doi:10.1051/ps:2007022

Cited by 6 publications

(4 citation statements)

References 18 publications

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“…However, this follows easily from Theorem 8, since we know that X − R H is smoother than R H . This gives a significantly shorter proof of Theorem 2.1 in [AS07]. The non-Gaussian stable case also treated in [AS07] cannot be handled this way due to the absence of the decorrelation inequality.…”

Section: Entropy Numbers Of Operatorsmentioning

confidence: 99%

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Path regularity of Gaussian processes via small deviations

Aurzada

2009

Preprint

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Section: Entropy Numbers Of Operatorsmentioning

confidence: 99%

“…where B is Brownian motion and f is a smooth function except possibly at zero, where we assume f (x) = x H−1/2 g(x) with H > 0 and a function g with g(0) = 1. This setup was studied in [AS07]. The goal is to show that X has the same small deviation order (and path regularity) as a Riemann-Liouville process R H .…”

Section: Entropy Numbers Of Operatorsmentioning

confidence: 99%

Path regularity of Gaussian processes via small deviations

Aurzada

2009

Preprint

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“…Remark. The results of Aurzada and Simon (2007) can also be improved to a larger range of H when considering L q -norms, q < ∞, by the use of Proposition 7.5.…”

Section: Riemann-liouville Processesmentioning

confidence: 99%

Small deviations of stable processes and entropy of the associated random operators

Аурзада¹,

Lifshits²,

Linde³

2009

Bernoulli

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We investigate the relation between the small deviation problem for a symmetric α-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric α-stable random processes, including unbounded Riemann-Liouville processes, weighted Riemann-Liouville processes and the (d-dimensional) α-stable sheet. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 4, 1305-1334. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2009 ISI/BS 1306 F. Aurzada, M. Lifshits and W. LindeIn this case, we say that X is generated by the operator u. This approach is very useful for investigating symmetric α-stable random processes with paths in E ′ . We refer to Section 5 of Li and Linde (2004) or Section 7.1.1 below for a discussion of how all natural examples of SαS processes fit into this framework. For example, if u from L p [0, 1] to L α [0, 1] is defined by (uf )(t) :=

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“…For Gaussian processes, such as fBm (β = 1), the small ball problem has been studied extensively [13], and exponential decay is typical. But there are also many works studying small ball probabilities for non-Gaussian processes; see, e.g., [1,2] and the references therein. We refer to [11,18] for other examples of processes with the small ball rate ε 2 of ggBm.…”

Section: Introductionmentioning

confidence: 99%

Small ball probabilities and large deviations for grey Brownian motion

Gerhold¹

2023

Preprint

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We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.

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Small ball probabilities for stable convolutions

Abstract: Mathematics Subject Classification. 60F99, 60G15, 60G20, 60G52.

Cited by 6 publications

References 18 publications

Path regularity of Gaussian processes via small deviations

Path regularity of Gaussian processes via small deviations

Small deviations of stable processes and entropy of the associated random operators

Small ball probabilities and large deviations for grey Brownian motion

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