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In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-It^o expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of stochastic processes.
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This article is devoted to studying the non-commutative Poisson boundary associated with $Big(Bbig(mathcal{F}(mathcal{H})big), P_{omega}Big)$ where $mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $dim mathcal{H} > 1$, with an orthonormal basis $mathcal{E}$, $Bbig(mathcal{F}(mathcal{H})big)$ is the algebra of bounded linear operators on the full Fock space $mathcal{F}(mathcal{H})$ defined over $mathcal{H}$, $omega = {omega_e : e in mathcal{E} }$ is a sequence of positive real numbers such that $sum_e omega_e = 1$ and $P_{omega}$ is the Markov operator on $Bbig(mathcal{F}(mathcal{H})big)$ defined by begin{align*} P_{omega}(x) = sum_{e in mathcal{E}} omega_e l_e^* x l_e, x in Bbig(mathcal{F}(mathcal{H})big), end{align*} where, for $e in mathcal{E}$, $l_e$ denotes the left creation operator associated with $e$. The non-commutative Poisson boundary associated with $Big(Bbig(mathcal{F}(mathcal{H})big), P_{omega}Big)$ turns out to be an injective factor of type $III$ for any choice of $omega$. Moreover, if $mathcal{H}$ is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes $S$-invarinat and curiously they are type $III _{lambda }$ factors with $lambda$ belonging to a certain small class of algebraic numbers.
We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Balls cube slicing inequality.
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Ruschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Ruschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitati
In this paper, we introduce the Fock space over $mathbb{C}^{infty}$ and obtain an isomorphism between the Fock space over $mathbb{C}^{infty}$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in cite{coburn1985}. As a physical application, we study the Gibbs state.
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