Digit frequencies and Bernoulli convolutions

It is well known that when β$\beta$ is a Pisot number, the corresponding Bernoulli convolution _νβ${\nu }_{\beta }$ has Hausdorff dimension less than 1$1$, i.e. that there exists a set _Aβ$A_{\beta }$ with _νβ(_Aβ)=1${\nu }_{\beta }\left(A_{\beta }\right)=1$ and _dimH(_Aβ)<1${dim}_H\left(A_{\beta }\right)<1$. We show explicitly how to construct for each Pisot number β$\beta$ such a set _Aβ$A_{\beta }$.     

Bernoulli thinning of a Markov renewal process

A Bernoulli thinning of a Markov renewal process is investigated. The properties of the thinned process are considered and are related to the properties of the original process. The parameters, moments and equilibrium of the thinned process are determined in terms of the parameters defining the underlying Markov renewal process. Results are illustrated by examples. © 1998 John Wiley & Sons, Ltd.     

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersley’s critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersley’s critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet.Most results of this paper are announced in Okamura (2016) [24] without proofs. This paper gives full details of them.     

On the Asymptotic Behavior of the Storage Process Fed by a Markov Modulated Brownian Motion

Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval [0, t] grows asymptotically like log t as t→∞.     

Density averaging for Markov processes and the invariant measures problem

A necessary and sufficient condition is given for the existence of a finite invariant measure equivalent to a given reference measure for a discrete time, general state Markov process. The condition is an extension of one given by D. Maharam in the deterministic case and involves an averaging method (called by Maraham ‘density averaging’) applied to the Radon-Nikodym derivatives with respect to the reference measure of the usual sequence of measures induced by the Markov process acting on the fixed reference     

Convergence of Markov Chains in Information Divergence

Information theoretic methods are used to prove convergence in information divergence of reversible Markov chains. Also some ergodic theorems for information divergence are proved.      

Set-valved Markov Processes and Their Representation Theorems

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Structural properties of Markov modulated revenue management problems

The admission decision is one of the fundamental categories of demand-management decisions. In the dynamic model of the single-resource capacity control problem, the distribution of demand does not explicitly depend on external conditions. However, in reality, demand may depend on the current external environment which represents the prevailing economic, financial, social or other factors that affect customer behavior. We formulate a Markov Decision Process (MDP) to maximize expected revenues over a finite horizon that explicitly models the current environment. We derive some structural results of the optimal admission policy, including the existence of an environment-dependent thresholds and a comparison of threshold levels in different environments. We also present some computational results which illustrate these structural properties. Finally, we extend some of the results to a related dynamic pricing formulation.     

Generation of Bernoulli processes

We consider five different algorithms for generating Bernoulli processes and discuss their efficiency. Simulation results are presented as well.     

A Markov modulated Poisson model for software reliability

In this paper, we consider a latent Markov process governing the intensity rate of a Poisson process model for software failures. The latent process enables us to infer performance of the debugging operations over time and allows us to deal with the imperfect debugging scenario. We develop the Bayesian inference for the model and also introduce a method to infer the unknown dimension of the Markov process. We illustrate the implementation of our model and the Bayesian approach by using actual software failure data.     

A queue with a pair of instantaneous independent bernoulli feedback processes

In this paper we are concerned with several random processes that occur in M/G₂/l queue with instantaneous feedback in which the feedback decision process is a pair of independent Bernoulli processes. The stationary distribution of the output process has been obtained. Results for particular queues with feedback and without feedback are obtained. Some operating characteristics are derived for this queue. Some interesting results are obtained for departure processes. Optimum service rate is obtained. Numerical examples are provided to test the feasibility of the queueing model     

高阶Bernoulli数的递推公式

本文得到了高阶 Bernoulli数的若干递推公式 ,这些公式不仅结构精美 ,递推关系鲜明 ,而且便于应用     

An identity of symmetry for the Bernoulli polynomials

The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials. It turns out that the recurrence relation and multiplication theorem for the Bernoulli polynomials which discussed in [F.T. Howard, Application of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995) 157-172], as well as a relation of symmetry between the power sum polynomials and the Bernoulli numbers developed in [H.J.H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001) 258-261], are all special cases of our results.     

TheM/GI/1 Bernoulli feedback queue with vacations

Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.This work was supported in part by NSF Grant No. DDM-8913658.     

Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

This paper describes an approach to generalized Bernoulli polynomials in higher dimensions by using Clifford algebras. Due to the fact that the obtained Bernoulli polynomials are special hypercomplex holomorphic (monogenic) functions in the sense of Clifford Analysis, they have properties very similar to those of the classical polynomials. Hypercomplex Pascal and Bernoulli matrices are defined and studied, thereby generalizing results recently obtained by Zhang and Wang (Z. Zhang, J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math. 154 (11) (2006) 1622-1632).     

On Miki's identity for Bernoulli numbers

We give a short proof of Miki's identity for Bernoulli numbers,

     

A Bernoulli problem with non-constant gradient boundary constraint

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.     

Ergodic and adaptive control of hidden Markov models

A partially observed stochastic system is described by a discrete time pair of Markov processes. The observed state process has a transition probability that is controlled and depends on a hidden Markov process that also can be controlled. The hidden Markov process is completely observed in a closed set, which in particular can be the empty set and only observed through the other process in the complement of this closed set. An ergodic control problem is solved by a vanishing discount approach. In the case when the transition operators for the observed state process and the hidden Markov process depend on a parameter and the closed set, where the hidden Markov process is completely observed, is nonempty and recurrent an adaptive control is constructed based on this family of estimates that is almost optimal.