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We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and…
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial…
The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0,1) random variables. Tribble [Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences (2007)…
A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle…
We consider two different stationary random processes whose probability distributions are very close and indistinguishable by standard tests for large but limited statistics. Yet we demonstrate that these processes can be reliably…
The probability distribution of the conductance p(g) of disordered 2d and 3d systems is calculated by transfer matrix techniques. As expected, p(g) is Gaussian for extended states while for localized states it is log-normal. We find that at…
We propose a generalization of classical statistical mechanics which describes the behavior of dissipative systems placed in contact with a heat bath. In contrast to conventional statistical mechanics, which assigns probabilities to the…
We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations…
We prove a conjecture of Haglund which can be seen as an extension of the equidistribution of the inversion number and the major index over permutations to ordered set partitions. Haglund's conjecture implicitly defines two statistics on…
Phase-type distribution has been an important probabilistic tool in the analysis of complex stochastic system evolution. It was introduced by Neuts \cite{Neuts1975} in 1975. The model describes the lifetime distribution of a finite-state…
We derive some key extremal features for $k$th order Markov chains that can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a…
We analyze certain stationary fields with linear regressions and quadratic conditional variances. This classic probabilistic problem leads somewhat unexpectedly to stationary Markov processes closely tied to non-commutative probability…
We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle…
Switching ARMA models greatly enhance the standard linear models to the extent that different ARMA model is allowed in a different regime, and the regime switching is typically assumed a Markov chain on the finite states of potential…
We propose a procedure to reconstruct characteristic features of an unknown stochastic system from the long-time full counting statistics of some of the system's transitions that are monitored by a detector. The full counting statistics is…
We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative…
In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In…
In this paper we consider the problem of computing the stationary distribution of nearly completely decomposable Markov processes, a well-established area in the classical theory of Markov processes with broad applications in the design,…
In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior…
Graphical Models have various applications in science and engineering which include physics, bioinformatics, telecommunication and etc. Usage of graphical models needs complex computations in order to evaluation of marginal functions,so…