Stochastic analysis of Bernoulli processes

Privault, Nicolas

doi:10.1214/08-ps139

Cited by 65 publications

(77 citation statements)

References 20 publications

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“…According e.g. to [31,32], the gradient operator admits the following representation. Let ω = (ω 1 , ω 2 , .…”

Section: Discrete Malliavin Calculus and A New Chain Rulementioning

confidence: 99%

“…With respect to [31,32], note that we choose to add a minus in the right-hand side of (2.18), in order to facilitate the connection with the paper [22]. One crucial relation between the operators δ, D and L is that…”

Section: Discrete Malliavin Calculus and A New Chain Rulementioning

confidence: 99%

“…random variables, such that P (X 1 = 1) = P (X 1 = −1) = 1/2. A full treatment of the discrete version of Malliavin calculus adopted in this paper can be found in [31] or [32]; Section 2.5 below provides a short introduction to the main objects and results.…”

Section: Introductionmentioning

confidence: 99%

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Stein's Method and Stochastic Analysis of Rademacher Functionals

Reinert

Nourdin

Peccati

2010

Electron. J. Probab.

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We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.

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“…According e.g. to [31,32], the gradient operator admits the following representation. Let ω = (ω 1 , ω 2 , .…”

Section: Discrete Malliavin Calculus and A New Chain Rulementioning

confidence: 99%

Section: Discrete Malliavin Calculus and A New Chain Rulementioning

confidence: 99%

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Stein's Method and Stochastic Analysis of Rademacher Functionals

Reinert

Nourdin

Peccati

2010

Electron. J. Probab.

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“…Discrete-time normal martingales [13] also play an important role in many theoretical and applied fields. For example, the classical random walk is just such a discretetime normal martingale [14,15].…”

Section: Introductionmentioning

confidence: 99%

Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

Chen

2018

Journal of Function Spaces

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We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales . We first apply the 2D-Fock transform for operators from the testing functional space S( ) to the generalized functional space S * ( ) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales . And also we prove the continuity and continuous dependence on initial values of the solution.

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“…It turns out [6] that the generalized functional space S * ( ) can accommodate many quantities of theoretical interest that cannot be covered by L 2 ( ). In this paper, we would like to extend the Clark-Ocone formula (1) to the generalized functionals of . More precisely, we would like to establish a Clark-Ocone formula for all elements of S * ( ).…”

Section: Introductionmentioning

confidence: 99%

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

Wang

Lin

Huang

2018

Journal of Function Spaces

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The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discretetime normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of . Then, we use the Fock transform to define some fundamental operators on generalized functionals of and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of .

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Stochastic analysis of Bernoulli processes

Cited by 65 publications

References 20 publications

Stein's Method and Stochastic Analysis of Rademacher Functionals

Stein's Method and Stochastic Analysis of Rademacher Functionals

Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

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