106 research outputs found
Differentiation Theory over Infinite-Dimensional Banach Spaces
No full textIn this paper we study, for any positive integer and for any subset\ \ of \QTR{bf}{N}^{\ast }, the Banach space of the bounded real sequences , and a measure over \left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) that generalizes the -dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables' formula for the integration of the measurable real functions on \left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) . This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms
Change of variables' formula for the integration of the measurable real functions over infinite-dimensional Banach spaces
No full textIn this paper we study, for any subset of and for
any strictly positive integer , the Banach space of the bounded
real sequences , and a measure over
that generalizes the
-dimensional Lebesgue one. Moreover, we recall the main results about the
differentiation theory over . The main result of our paper is a change
of variables' formula for the integration of the measurable real functions on
. This change of variables
is defined by some functions over an open subset of , with values on
, called -general, with properties that
generalize the analogous ones of the finite-dimensional diffeomorphisms
Generating uniform random vectors in (Z_p)^k: the general case
No full textWe consider the rate of convergence of the Markov chain Xn+1 = AXn + Bn (modp), where A is an integer matrix with nonzero eigenvalues, and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Qk invariant under A. If |λi | =1 for all eigenvalues λi of A, then n = O((ln p)2) steps are sufficient and n = O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1| = 1 and |λi | > 1 for all i = 1, then O(p2) steps are necessary and sufficient
Theory of the (m,σ)-general functions over infinite-dimensional Banach spaces
No full textIn this paper, we introduce some functions, called (m, σ)-general, that generalize the (m, σ)-standard functions and are defined in the infinite-dimensional Banach space E_I of the bounded real sequences {xn}n∈I, for some subset I of N*. Moreover, we recall the main results about the differentiation theory over E_I, and we expose some properties of the (m, σ)-general functions. Finally, we study the linear (m, σ)-general functions, by introducing a theory that generalizes the standard theory of the m × m matrices
Convergence in total variation of an affine random recursion in [0,p)^k to a uniform random vector
No full textWe study the rate of convergence of the Markov chain (mod ), where is an integer matrix with nonzero eigenvalues, is real and positive, and is a sequence of independent and identically distributed real random vectors. With some hypotheses on the law of , the sequence converges to a random vector uniformly distributed in . The rate of
convergence is geometric and depends on , , , and the distribution of . Moreover, if has an eigenvalue that is a root of , then steps are necessary to have
sampling from a nearly uniform law
Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications
No full textIn this paper we study, for some subsets I of N^{∗}, the Banach space E of bounded real sequences {x_{n}}_{n∈I}. For any integer k, we introduce a measure over (E,B(E)) that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables formula for the integration
Infinite-dimensional Gaussian change of variables’ formula
No full textIn this paper, we study the Banach space ∞ of the bounded real sequences, and a
measure N(a, ) over (R∞
, B∞
) analogous to the finite-dimensional Gaussian law.
The main result of our paper is a change of variables’ formula for the integration,
with respect to N(a, ), of the measurable real functions on (E∞, B∞
(E∞)), where
E∞ is the separable Banach space of the convergent real sequences. This change of
variables is given by some (m, σ) functions, defined over a subset of E∞, with values
on E∞, with properties that generalize the analogous ones of the finite-dimensional
diffeomorphisms
Asymptotic behavior of an affine random recursion in (Z_p)^k defined by a matrix with an eigenvalue of size 1
No full textIn this paper we study the rate of convergence of the Markov chain XnC1 D AXn C Bn.mod p/, where A is an integer invertible matrix, and fBngn is a sequence of independent and identically distributed integer vectors. If A has an eigenvalue of size 1, then n D O.p2/ steps are necessary and sufficient to have Xn sampling from a nearly uniform distribution
Functionally Compatible Local Characteristics for the Local Specification of Priors in Graphical Models
No full textThe local specification of priors in non-decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..
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