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Register a new userWick order, spreadability and exchangeability for monotone commutation relations
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We exhibit a Hamel basis for the concrete $*$-algebra $mathfrak{M}_o$ associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure $C^*$-algebra $mathfrak{M} = overline{mathfrak{M}_o}$. Moreover, we determine the structure of spreadable and exchangeable monotone stochastic processes using their correspondence with sp-reading invariant and symmetric monotone states, respectively.
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We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form [ L_{e_i}L_{f_j} = sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}] where $u= (u_{i,j,k,l})$ is an $n
m times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.
Exchangeability -- in which the distribution of an infinite sequence is invariant to reorderings of its elements -- implies the existence of a simple conditional independence structure that may be leveraged in the design of probabilistic models, efficient inference algorithms, and randomization-based testing procedures. In practice, however, this assumption is too strong an idealization; the distribution typically fails to be exactly invariant to permutations and de Finettis representation theory does not apply. Thus there is the need for a distributional assumption that is both weak enough to hold in practice, and strong enough to guarantee a useful underlying representation. We introduce a relaxed notion of local exchangeability -- where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. We thereby show that de Finettis theorem is robust to perturbation and provide further justification for the Bayesian modelling approach. Using this probabilistic result, we develop three novel statistical procedures for (1) estimating the underlying process via local empirical measures, (2) testing via local randomization, and (3) estimating the canonical premetric of local exchangeability. These three procedures extend the applicability of previous exchangeability-based methods without sacrificing rigorous statistical guarantees. The paper concludes with examples of popular statistical models that exhibit local exchangeability.
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescus fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescus fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.
We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have been discussed for infinite matrix size. Moreover we derive the joint probability densities of the eigenvalues. To illustrate our results we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights we show that the product of complex random matrices yield a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.
We investigate generalizations of the Cramer 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.
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