This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers.Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states.A description is given of all invariant Riemannian metrics on manifolds of sectorial states.The equations of the geodesics for the entire family of invariant linear connections A = ~A, yEIR, are integrated on sets of classical probability distributions.A description is given of the projective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor T(7 -I), with the Riemann-Christoffel curvature tensor.I. The collection of all probability measures P(.) on a measurable space (~, ~) of elementary outcomes ~ is a convex subset of the semi-ordered linear space of all countably additive charges R, i.e., measures of bounded variation on (~,~). Compatible with the semiorder relation of this space is the natural norm -the total variation of the charge
In 1934 A. N. Kolmogorov proved the following Theorem. I/(t) is a separable (see [1]) stochastic process, (1) Ml(tl)-(t)i < clq-t,.I+, where p > O, r > 0 and C is a constant independent o/ t, then the trajectories o/ the process are continuous with probability 1% A generalization of this theorem is the following proposition which was suggested to the author by A. N. Kolmogorov" Theorem 1. I/ (t) is a separable stochastic process, 0 < _ 1, andand C is a constant independent o/t, then the trajectories o/ the process have no discontinuities o/ the second kind with probability 1. PROOF. It is easy to show that if (t) satisfies (2), any denumerable set which is dense in the rel line may be tken as the set relative to which the process is separable, for example, the set 0 of rational binary fractions. By A+ we will mean the event [(k/2) --((k+ 1)/2n)[ < Ln, where exp [--(r log 2)/2(p + q)], L+q C. Further, we introduce the events A_tO A+, the events Dn= B, and the event D k=l It follows easily from inequality (2) hat P(} < -n(+q)2-n(l+r); P{Dn} < n/(1--), where 2 -< 1, and finally P(D} 1. I remains o show hat he set D has no trajectory with discontinuities of the second kind which is separable relative Lemma 1. I/ x(t) is a /unction separable relative to F. and x(t) A+t D n+l, thenmax ()--< +1 << () max (t)--XJ < 1"
<
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.