Journal•ISSN: 0178-8051

# Probability Theory and Related Fields

Springer Science+Business Media

About: Probability Theory and Related Fields is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Probability theory & Random walk. It has an ISSN identifier of 0178-8051. Over the lifetime, 4708 publications have been published receiving 188946 citations. The journal is also known as: Probability theory (Internet) & Probability theory (Print).

##### Papers published on a yearly basis

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1,629 citations

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TL;DR: In this article, a probability density on an interval I, finite or infinite, including its finite endpoints, if any; and f vanishes outside of I. To define this object, choose a reference point xosI and a cell width h.

Abstract: Let f be a probability density on an interval I, finite or infinite: I includes its finite endpoints, if any; and f vanishes outside of I. Let X1, . . . ,X k be independent random variables, with common density f The empirical histogram for the X's is often used to estimate f To define this object, choose a reference point xosI and a cell width h. Let Nj be the number of X's falling in the j th class interval:

1,336 citations

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TL;DR: In this article, the authors introduced a new construction for the pair S¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ n�, T¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ n>>\s, and proved that if X>>\s has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦S>>\s n� -T� n� nၡ 1/4(log n) 1/1(log log n)1/4) with probability one.

Abstract: Let S
n
=X
1+X
2+⋯+X
n
be the sum of i.i.d.r.v.-s, EX
1=0, EX
1
2
=1, and let T
n
= Y
1+Y
2+⋯+Y
n
be the sum of independent standard normal variables. Strassen proved in [14] that if X
1 has a finite fourth moment, then there are appropriate versions of S
n
and T
n
(which, of course, are far from being independent) such that ¦S
n
-T
n
¦=O(n
1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bartfai [1] indicates that even if X
1 has a finite moment generating function, the best possible bound for any version of S
n
, T
n
is O(log n). In this paper we introduce a new construction for the pair S
n
, T
n
, and prove that if X
1 has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦S
n
-T
n
¦=O(log n) with probability one for the constructed S
n
, T
n
. Our method will be applicable for the approximation of sample DF., too.

1,190 citations

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TL;DR: In this article, a criterion for existence of global random attractors for RDS is established and the existence of invariant Markov measures supported by the random attractor is proved for SPDE, which yields invariant measures for the associated Markov semigroup.

Abstract: A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.

902 citations

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TL;DR: It is shown that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve, which quantifies the trade-off among the candidate models between the approximation error and parameter dimension relative to sample size.

Abstract: Performance bounds for criteria for model selection are devel- oped using recent theory for sieves. The model selection criteria are based on an empirical loss or contrast function with an added penalty term moti- vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob- servations. Most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function. We show that the quadratic risk of the minimum penal- ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve. This accuracy index quanties the trade-off among the candidate models between the approximation error and parameter dimension relative to sample size. If we choose a list of models which exhibit good approximation prop- erties with respect to different classes of smoothness, the estimator can be simultaneously minimax rate optimal in each of those classes. This is what is usually called adaptation. The type of classes of smoothness in which one gets adaptation depends heavily on the list of models. If too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously, it may happen that the estimator is only approx-

801 citations