On Independence and Conditioning On Wiener Space

Üstünel, Ali Süleyman; Zakai, Moshe

doi:10.1214/aop/1176991164

Cited by 43 publications

(50 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…q 1 +q 2 −2 + = 0, see [UZ89]. Above, f ⊗ 1 g is an instance of the contraction between f and g which, in general, is defined for 0 ≤ ≤ q 1 ∧ q 2 := min(q 1 , q 2 ), by…”

Section: Multiple Wiener Integralsmentioning

confidence: 99%

Asymptotic Cramér Type Decomposition for Wiener and Wigner Integrals

Bourguin

Breton

2013

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

View full text Add to dashboard Cite

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

show abstract

“…q 1 +q 2 −2 + = 0, see [UZ89]. Above, f ⊗ 1 g is an instance of the contraction between f and g which, in general, is defined for 0 ≤ ≤ q 1 ∧ q 2 := min(q 1 , q 2 ), by…”

Section: Multiple Wiener Integralsmentioning

confidence: 99%

Asymptotic Cramér Type Decomposition for Wiener and Wigner Integrals

Bourguin

Breton

2013

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

View full text Add to dashboard Cite

show abstract

“…We use a result byÜstunel-Zakai [41] (see also Kallenberg [20]): two multiple Wiener-Itô integrals with respect to the standard Wiener process I n (f ) and…”

Section: Remarkmentioning

confidence: 99%

Analysis of the Rosenblatt process

2008

View full text Add to dashboard Cite

Abstract. We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.Mathematics Subject Classification. 60G12, 60G15, 60H05, 60H07.

show abstract

“…By making use of the standard properties of multiple Wiener-Ito integrals, Ustunel and Zakai [10] have shown that two homogeneous MWIs are independent if and only if their squares are uncorrelated. An extension of their argument, which we give here, provides a corresponding criterion for asymptotic independence of two sequences of MWIs:…”

Section: Asymptotic Independence Of Homogeneous Mwismentioning

confidence: 99%

“…The MWIs treated by Ustunel and Zakai [10] are real integrals of real integrands as in Ito [2]. The same ideas also apply to the complex MWIs of Ito [4], and since it is the latter integrals which arise in the spectral theory for stationary Gaussian and subordinated processes, here we treat only the complex case.…”

Section: Asymptotic Independence Of Homogeneous Mwismentioning

confidence: 99%

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

Slud

Chambers

1996

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Link to this article: http://journals.cambridge.org/abstract_S0143385700010191How to cite this article: Eric Slud and Daniel Chambers (1996). Mixing for stationary processes with nite-order multiple Wiener-Itô integral representation.Abstract. Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Ito integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y,, for sums of MWIs of order < 3, and for a large class of square-integrable infinite sums Y, of MWIs, mixing holds if and only if K, 2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

show abstract

On Independence and Conditioning On Wiener Space

Cited by 43 publications

References 7 publications

Asymptotic Cramér Type Decomposition for Wiener and Wigner Integrals

Asymptotic Cramér Type Decomposition for Wiener and Wigner Integrals

Analysis of the Rosenblatt process

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

Contact Info

Product

Resources

About