Convergence In Total Variation Research Articles

Shift-coupling in continuous time

The result linking shift-coupling to time-average total variation convergence and to the invariant σ-field is extended to continuous time and an analogous result established linking e-couplings to smooth total variation convergence and to a smooth tail σ-field. Shift- and e-coupling inequalities are presented.

Stochastic annealing for nearest-neighbour point processes with application to object recognition

We study convergence in total variation of non-stationary Markov chains in continuous time and apply the results to the image analysis problem of object recognition. The input is a grey-scale or binary image and the desired output is a graphical pattern in continuous space, such as a list of geometric objects or a line drawing. The natural prior models are Markov point processes found in stochastic geometry. We construct well-defined spatial birth-and-death processes that converge weakly to the posterior distribution. A simulated annealing algorithm involving a sequence of spatial birth-and-death processes is developed and shown to converge in total variation to a uniform distribution on the set of posterior mode solutions. The method is demonstrated on a tame example.

Stochastic annealing for nearest-neighbour point processes with application to object recognition

We study convergence in total variation of non-stationary Markov chains in continuous time and apply the results to the image analysis problem of object recognition. The input is a grey-scale or binary image and the desired output is a graphical pattern in continuous space, such as a list of geometric objects or a line drawing. The natural prior models are Markov point processes found in stochastic geometry. We construct well-defined spatial birth-and-death processes that converge weakly to the posterior distribution. A simulated annealing algorithm involving a sequence of spatial birth-and-death processes is developed and shown to converge in total variation to a uniform distribution on the set of posterior mode solutions. The method is demonstrated on a tame example.

Uniform Cesaro limit theorems for synchronous processes with applications to queues

Let X={ X( t): t⩾0} be a positive recurrent synchronous process (PRS), that is, a process for which there exists an increasing sequence of random times τ={ τ( k)} such that for each k the distribution of θ τ(k)X = def {X(t+τ(k)):t⩾0} is the same and the cycle lengths T n = def τ(n+1)−τ(n) have finite first moment. Such processes (in general) do not converge to steady-state weakly (or in total variation) even when regularity conditions are placed on the cycles (such as non-lattice, spread-out, or mixing). Nonetheless, in the present paper we first show that the distributions of { θ s X: s > 0} are tight in the function space D(0,∞) . Then we investigate conditions under which the Cesaro averaged functionals μ ̄ i = def ( 1 t )∫ t 0E(ƒ(θ sX)) ds converge uniformly (over a class of functions) to π(ƒ), where π is the stationary distribution of X. We show that μ t(ƒ)→π(ƒ) uniformly over ƒ satisfying ‖ƒ‖ ∞⩽1 (total variation convergence). We also show that to obtain uniform convergence over all ƒ satisfying |ƒ|⩽g (gϵL + 1(π) fixed) requires placing further conditions on the PRS. This is in sharp contrast to both classical regenerative processes and discrete time Harris recurrent Markov chains (where renewal theory can be applied) where such uniform convergence holds without any further conditions. For continuous time positive Harris recurrent Markov processes (where renewal theory cannot be applied) we show that these further conditions are in fact automatically satisfied. In this context, applications to queueing models are given.

Distribution estimation consistent in total variation and in two types of information divergence

The problem of the nonparametric estimation of a probability distribution is considered from three viewpoints: the consistency in total variation, the consistency in information divergence, and consistency in reversed-order information divergence. These types of consistencies are relatively strong criteria of convergence, and a probability distribution cannot be consistently estimated in either type of convergence without any restrictions on the class of probability distributions allowed. Histogram-based estimators of distribution are presented which, under certain conditions, converge in total variation, in information divergence, and in reversed-order information divergence to the unknown probability distribution. Some a priori information about the true probability distribution is assumed in each case. As the concept of consistency in information divergence is stronger than that of convergence in total variation, additional assumptions are imposed in the cases of informational divergences. >

An application to probability laws of the noncontinuity of the inverse Radon transform

While any ƒ∈ L 1( R n ) is uniquely determined by its Radon transform, we show in this note that the inverse transform is not continuous on L ∞,1( S n-1 × R ) and provide a probabilistic interpretation: the Cramér—Wold device fails when convergence in total variation replaces weak convergence.

Notes on the stability of closed queueing networks

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.

Notes on the stability of closed queueing networks

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.

Corrigendum to "A Tool in Establishing Total Variation Convergence"

In Theorem 2.1 of [1] the measure ,u on the Borel-sets of a complete and seperable metric space X was assumed to be cr-finite. The a-finiteness condition has to be replaced by the condition that it is a Radon-measure. In ?4 we stated that the bounded continuous functions with support of finite diameter are dense in 11. This is not true in general, as can be seen by the following simple counterexample due to Professor Erik Thomas. Take X = [0, 1] and d(x, y) = Ixy. Define the a-finite measure /i = &o + EZ=i 61/n, where &, denotes the Dirac-measure in the point x. The continuous, integrable functions are not dense in iZ. For any f E C n Li we have f f df = f(0) + ?ZnL1 f(n-1) . From the continuity of f it follows that f(0) = 0. Define f* 1 by f* = I{o}. Note that f If f* dlu > 1 for any f C n 0 1. The problem arises because the point 0 does not have an open neighborhood of finite measure. The outer regularity is needed. So, we have to assume that it is a Radon-measure, which is cr-finite. In this case the bounded, continuous, integrable functions with support of finite diameter are dense in 121. A direct proof can easily be given, but it is also an easy consequence of a nice result (Proposition 2.1) of Thomas [2]. We wish to thank Professor E. Thomas for drawing our attention to his counterexample.

Corrigendum to: “A tool in establishing total variation convergence” [Proc. Amer. Math. Soc. 95 (1985), no. 4, 626–630; MR0810175 (87b:60005)

Corrigendum to: “A tool in establishing total variation convergence” [Proc. Amer. Math. Soc. 95 (1985), no. 4, 626–630; MR0810175 (87b:60005)

Rates of convergence to the stationary distribution for k-dimensional diffusion processes

Using a coupling technique, rates of convergence in total variation of certain k -dimensional diffusion processes to the stationary distribution are obtained. The results place assumptions only on the coefficients of the elliptic differential operator governing the diffusion.

Rates of convergence to the stationary distribution for k-dimensional diffusion processes

Using a coupling technique, rates of convergence in total variation of certain k -dimensional diffusion processes to the stationary distribution are obtained. The results place assumptions only on the coefficients of the elliptic differential operator governing the diffusion.

A Tool in Establishing Total Variation Convergence

Let ${X_0},{X_1},{X_2}, \ldots {\text { and }}{Y_0},{Y_1},{Y_2}, \ldots$ be sequences of random variables where ${X_n}$ and ${Y_n}$ are independent, $L{X_n} \to L{X_0}$ in total variation and $L{Y_n} \to L{Y_0}$ in distribution. For certain mappings $T$ sufficient conditions are given in order that $LT\left ( {{X_n},{Y_n}} \right ) \to LT\left ( {{X_0},{Y_0}} \right )$ in total variation. For example, if $\left ( {{{\mathbf {R}}^k},{B_k}} \right )$ is the outcome space of the ${X_n}$ and ${Y_n}$, and if $L{X_0}$ is absolutely continuous (with respect to Lebesgue measure), then $L\left ( {{X_n} + {Y_n}} \right ) \to L\left ( {{X_0} + {Y_0}} \right )$ in total variation.

A tool in establishing total variation convergence

Let X 0 , X 1 , X 2 , … and Y 0 , Y 1 , Y 2 , … {X_0},{X_1},{X_2}, \ldots {\text { and }}{Y_0},{Y_1},{Y_2}, \ldots be sequences of random variables where X n {X_n} and Y n {Y_n} are independent, L X n → L X 0 L{X_n} \to L{X_0} in total variation and L Y n → L Y 0 L{Y_n} \to L{Y_0} in distribution. For certain mappings T T sufficient conditions are given in order that L T ( X n , Y n ) → L T ( X 0 , Y 0 ) LT\left ( {{X_n},{Y_n}} \right ) \to LT\left ( {{X_0},{Y_0}} \right ) in total variation. For example, if ( R k , B k ) \left ( {{{\mathbf {R}}^k},{B_k}} \right ) is the outcome space of the X n {X_n} and Y n {Y_n} , and if L X 0 L{X_0} is absolutely continuous (with respect to Lebesgue measure), then L ( X n + Y n ) → L ( X 0 + Y 0 ) L\left ( {{X_n} + {Y_n}} \right ) \to L\left ( {{X_0} + {Y_0}} \right ) in total variation.

Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm

Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class $\mathscr{F}$ of functions are proved under several kinds of conditions on $\mathscr{F}$. These bounds are used to establish laws of the iterated logarithm for this supremum and to obtain rates of convergence in total variation for empirical processes on the integers.

The central limit theorem for Markov chains with the simultatneous total variation convergence of the chain.

The central limit theorem for Markov chains with the simultatneous total variation convergence of the chain.

Total Variation and Uniform Convergence

Total Variation and Uniform Convergence