arXiv subjects

Mathukumalli Vidyasagar

Publications and source records attributed to Mathukumalli Vidyasagar.

21 records · Page 2Linked to original sources

arXiv2013

An Elementary Derivation of the Large Deviation Rate Function for Finite State Markov Chains

Large deviation theory is a branch of probability theory that is devoted to a study of the "rate" at which empirical estimates of various quantities converge to their true values. The object of study in this paper is the rate at which estimates of the doublet frequencies of a Markov chain over a finite alphabet converge to their true values. In case the Markov process is actually an i.i.d.\ process, the rate function turns out to be the relative entropy (or Kullback-Leibler divergence) between the true and the estimated probability vectors. This result is a special case of a very general result known as Sanov's theorem and dates back to 1957. Moreover, since the introduction of the "method of types" by Csiszár and his co-workers during the 1980s, the proof of this version of Sanov's theorem has been "elementary," using some combinatorial arguments. However, when the i.i.d.\ process is replaced by a Markov process, the available proofs are far more complex. The main objective of this paper is therefore to present a first-principles derivation of the LDP for finite state Markov chains, using only simple combinatorial arguments (e.g.\ the method of types), thus gathering in one place various arguments and estimates that are scattered over the literature.

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arXiv2013

Mixing Coefficients Between Discrete and Real Random Variables: Computation and Properties

In this paper we study the problem of estimating the alpha-, beta- and phi-mixing coefficients between two random variables, that can either assume values in a finite set or the set of real numbers. In either case, explicit closed-form formulas for the beta-mixing coefficient are already known. Therefore for random variables assuming values in a finite set, our contributions are two-fold: (i) In the case of the alpha-mixing coefficient, we show that determining whether or not it exceeds a prespecified threshold is NP-complete, and provide efficiently computable upper and lower bounds. (ii) We derive an exact closed-form formula for the phi-mixing coefficient. Next, we prove analogs of the data-processing inequality from information theory for each of the three kinds of mixing coefficients. Then we move on to real-valued random variables, and show that by using percentile binning and allowing the number of bins to increase more slowly than the number of samples, we can generate empirical estimates that are consistent, i.e., converge to the true values as the number of samples approaches infinity.

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arXiv2011

A Metric Between Probability Distributions on Finite Sets of Different Cardinalities and Applications to Order Reduction

With increasing use of digital control it is natural to view control inputs and outputs as stochastic processes assuming values over finite alphabets rather than in a Euclidean space. As control over networks becomes increasingly common, data compression by reducing the size of the input and output alphabets without losing the fidelity of representation becomes relevant. This requires us to define a notion of distance between two stochastic processes assuming values in distinct sets, possibly of different cardinalities. If the two processes are i.i.d., then the problem becomes one of defining a metric between two probability distributions over distinct finite sets of possibly different cardinalities. This is the problem addressed in the present paper. A metric is defined in terms of a joint distribution on the product of the two sets, which has the two given distributions as its marginals, and has minimum entropy. Computing the metric exactly turns out to be NP-hard. Therefore an efficient greedy algorithm is presented for finding an upper bound on the distance. This problem also turns out to be NP-hard, so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation. Taken together, all the results presented here permit the approximation of an i.i.d. process over a set of large cardinality by another i.i.d. process over a set of smaller cardinality. In future work, attempts will be made to extend this work to Markov processes over finite sets.

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