Showing papers on "Gaussian measure published in 1975"
TL;DR: In this article, the authors show that the measures induced by (X 1 + ··· + X n ) n − 1 2 converge weakly to a Gaussian measure on C(S) under different conditions on X1, one of which consolidates and extends results of Strassen and Dudley, Gine, and Dudley.
122 citations
01 Jan 1975
TL;DR: In this paper, it is shown that for one (resp. two) dimensions, it is possible to define an infinite volume equilibrium state on the algebra generated by the difference variables (Lanford and Lebowitz, 1974).
Abstract: The equilibrium and non-equilibrium properties of regular harmonic crystals are well understood (Montroll, 1956, Lanford and Lebowitz, 1974). For them it is possible to define an infinite volume equilibrium state for dimensionality three and higher. This state is a Gaussian measure. For one (resp. two) dimensions it is not possible to define such a state because the mean square displacement of any particle diverges as |A| (resp. ln|A|) as |A| →∞, where |A|) is the volume of the system. In all dimensions, however, it is possible to define a state on the algebra generated by the difference variables (Lanford and Lebowitz, 1974).
40 citations
TL;DR: The main result of as discussed by the authors is that the topological support of an arbitrary Gaussian measure in a separable Banach space can be described by the notion of covariance operator of a probability measure.
Abstract: The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]). Let X be a real Banach space with topological dual X* and //bea probability measure on the σ-algebra generated by X*. We suppose that μ has the second order in weak sense, i.e. X* c L2(X,μ) (this restriction is necessary if we want to speak about an analogue of variance). Covariance operator R of measure μ is defined by the relation (see [1])
22 citations