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Richard C. Bradley

Publications and source records attributed to Richard C. Bradley.

18 recordsLinked to original sources
arXiv2026

On the central limit question for strictly stationary, reversible Markov chains

This paper will provide several classes of strictly stationary, countable-state, irreducible, aperiodic Markov chains that are reversible and have finite second moments, such that the central limit theorem fails to hold. The main purpose is to examine the extent to which, for the development of central limit theory for strictly stationary Markov chains (and functions of them) under the strong mixing and absolute regularity conditions, the property of reversibility (if it holds) can provide extra leverage. It is known, partly as a by-product of research done by Roberts, Rosenthal, and Tweedie in two papers in 1997 and 2001, that for the case of exponential mixing rates, reversibility provides notable extra leverage of that kind. In contrast, a class of counterexamples in a paper of Doukhan, Massart, and Rio in 1994 showed (implicitly) that for the case of power-type mixing rates, reversibility apparently provides almost no such extra leverage. Further perspective on that latter fact will be provided by some counterexamples in this paper. Other counterexamples here will (indirectly) provide some tentative, uncertain evidence for the possibility that for mixing rates that are ``between'' power-type and exponential (for example, sub-exponential), reversibility may in fact provide some small but nontrivial extra leverage.

math.PR↗
arXiv2024

On a boundary of the central limit theorem for strictly stationary, reversible Markov chains

Consider the class of (functions of) strictly stationary Markov chains in which (i) the second moments are finite and (ii) absolute regularity (beta-mixing) is satisfied with exponential mixing rate. For (functions of) Markov chains in that class that are also reversible, the central limit theorem holds, as a well known byproduct of results of Roberts, Rosenthal, and Tweedie in two papers in 1997 and 2001 involving reversible Markov chains. In contrast, for (functions of) Markov chains in that class that are not reversible, the central limit theorem may fail to hold, as is known from counterexamples, including ones with arbitrarily fast mixing rate (for absolute regularity). Here it will be shown that for Markov chains in that class that are reversible, the``borderline'' class of mixing rates (for absolute regularity) for the central limit theorem is in fact exponential. That will be shown here with a class of counterexamples: strictly stationary, countable-state Markov chains that are reversible, have finite second moments, and satisfy absolute regularity with mixing rates that can be arbitrarily close to (but not quite) exponential, but fail to satisfy the central limit theorem.

math.PR↗
arXiv2022

On some possible combinations of mixing rates for strictly stationary, reversible Markov chains

A class of examples is constructed to show that for strictly stationary Markov chains that are reversible, the simultaneous mixing rates for the $ρ$-mixing and strong mixing ($α$-mixing) conditions can be fairly arbitrary, within certain unavoidable tight restrictions. The examples constructed here have the added property that the mixing rate for the absolute regularity ($β$-mixing) condition is within a constant factor of that for strong mixing.

math.PR↗
arXiv2019

An exposition of some basic features of strictly stationary, reversible Markov chains

It has been well known for some time that for strictly stationary Markov chains that are ``reversible'', that special symmetry provides special extra features in the mathematical theory. This paper here is primarily a purely expository review of some of the basic features of that special theory. The mathematical techniques employed in this review are relatively gentle, involving only some basic measure-theoretic probability theory.

math.PR↗
arXiv2016

A "cousin" of a theorem of Csáki and Fischer

A 1963 theorem of P. Csáki and J. Fischer deals with the "maximal correlation coefficient" in the context of independent pairs of $σ$-fields on a probability space. Here a somewhat restricted "cousin" of their result is presented for the same context, but involving in part an analogous measure of dependence based only on correlations of indicator functions. It was first proved by the author in an unpublished 1978 Ph.D. thesis. An example is constructed to show a limitation of this "cousin". Also, this "cousin" is used to trivially embellish a very sharp 2013 example of R. Peyre in connection with the comparison of these two measures of dependence.

math.PR↗
arXiv2015

A Central Limit Theorem for Non-Stationary Strongly Mixing Random Fields

In this paper we extend a central limit theorem of Peligrad for uniformly strong mixing random fields satisfying the Lindeberg condition in the absence of stationarity property. More precisely, we study the asymptotic normality of the partial sums of uniformly $α$-mixing non-stationary random fields satisfying the Lindeberg condition, in the presence of an extra dependence assumption involving maximal correlations.

math.PR↗
arXiv2014

On a central limit theorem for shrunken weakly dependent random variables

A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For independent, identically distributed random variables, this result was proved earlier by Housworth and Shao.

math.PR↗
arXiv2014

On Mixing Properties of Reversible Markov Chains

It is well known that for a strictly stationary, reversible, Harris recurrent Markov chain, the $ρ$-mixing condition is equivalent to geometric ergodicity and to a "spectral gap" condition. In this note, it will be shown with an example that for that class of Markov chains, the "interlaced" variant of the $ρ$-mixing condition fails to be equivalent to those conditions.

math.PR↗
arXiv2014

Snarks from a Kaszonyi perspective: a survey

This is a survey or exposition of a particular collection of results and open problems involving snarks --- simple "cubic" (3-valent) graphs for which, for nontrivial reasons, the edges cannot be 3-colored. The results and problems here are rooted in a series of papers by Laszlo Kaszonyi that were published in the early 1970s. The problems posed in this survey paper can be tackled without too much specialized mathematical preparation, and in particular seem well suited for interested undergraduate mathematics students to pursue as independent research projects. This survey paper is intended to facilitate research on these problems.

math.CO↗
arXiv2014

The strong mixing and the operator-selfdecomposability properties

For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the possible limit distributions are precisely the operator-selfdecomposable laws.

math.PR↗
arXiv2012

On a "replicating character string" model

In a paper of Chaudhuri and Dasgupta published in 2006, a certain stochastic model for "replicating character strings" (such as in DNA sequences) was studied. In their model, a random "input" sequence was subjected to random mutations, insertions, and deletions, resulting in a random "output" sequence. In this note, their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that under certain conditions, strict stationarity of the "input" sequence would be preserved by the "output" sequence, and they proved a similar "preservation" result for the property of strong mixing with exponential mixing rate. In our setup, we shall in spirit slightly extend their "preservation of stationarity" result, and also prove a "preservation" result for the property of absolute regularity with summable mixing rate.

math.PR↗
arXiv2011

On possible mixing rates for some strong mixing conditions for N-tuplewise independent random fields

For a given pair of positive integers $d$ and $N$ with $N \geq 2$, for strictly stationary random fields that are indexed by the $d$-dimensional integer lattice and satisfy $N$-tuplewise independence, the dependence coefficients associated with the $ρ$-, $ρ'$-, and $ρ^*$-mixing conditions can decay together at an arbitrary rate. Another, closely related result is also established. In particular, these constructions provide classes of examples pertinent to limit theory for random fields that involve such mixing conditions together with certain types of "extra" assumptions on the marginal and bivariate (or $N$-variate) distributions.

math.PR↗
arXiv2011

On the behavior of the covariance matrices in a multivariate central limit theorem under some mixing conditions

In a paper that appeared in 2010, C. Tone proved a multivariate central limit theorem for some strictly stationary random fields of random vectors satisfying certain mixing conditions. The "normalization" of a given "partial sum" (or "block sum") involved matrix multiplication by a "standard -1/2 power" of its covariance matrix (a symmetric, positive definite matrix), and the limiting multivariate normal distribution had the identity matrix as its covariance matrix. The mixing assumptions in Tone's result implicitly imposed an upper bound on the ratios of the largest to the smallest eigenvalues in the covariance matrices of the partial sums. The purpose of this note is to show that in Tone's result, for the entire collection of the covariance matrices of the partial sums, there is essentially no other restriction on the relative magnitudes of the eigenvalues or on the (orthogonal) directions of the corresponding eigenvectors. For simplicity, the example given in this note will involve just random sequences, not the broader context of random fields.

math.PR↗
arXiv2009

A strictly stationary, "causal," 5-tuplewise independent counterexample to the central limit theorem

A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent, (iii) the sequence is "causal" in a certain sense, (iv) the sequence has a trivial double tail sigma-field, and (v) regardless of the normalization used, the partial sums do not converge to a (nondegenerate) normal law. The example has some features in common with a recent construction (for an arbitrary fixed positive integer N), by Alexander Pruss and the author, of a strictly stationary N-tuplewise independent counterexample to the central limit theorem.

math.PR↗
arXiv2008

A strictly stationary, N-tuplewise independent counterexample to the central limit theorem

For an arbitrary integer N that is at least 2, this paper gives a construction of a strictly stationary, N-tuplewise independent sequence of (non-degenerate) bounded random variables such that the Central Limit Theorem fails to hold. The sequence is in part an adaptation of a non-stationary example with similar properties constructed by one of the authors (ARP) in a paper published in 1998.

math.PR↗