scholarly journals The Difference between Letters and a Martin Kernel of a Modulo 5 Markov Chain

2002 ◽  
Vol 28 (1) ◽  
pp. 82-106 ◽  
Author(s):  
Atsushi Imai
Keyword(s):  
Markov Chain ◽  
Martin Kernel ◽  
The Difference

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals The Limit Theorems for Function of Markov Chains in the Environment of Single Infinite Markovian Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Zhanfeng Li ◽  
Min Huang ◽  
Xiaohua Meng ◽  
Xiangyu Ge
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Strong Convergence ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Strong Law ◽  
Large Numbers ◽  
Finite State ◽  
The Difference ◽  
Markovian Systems

This paper is intended to study the limit theorem of Markov chain function in the environment of single infinite Markovian systems. Moreover, the problem of the strong law of large numbers in the infinite environment is presented by means of constructing martingale differential sequence for the measurement under some different sufficient conditions. If the sequence of even functions gnx,n≥0 satisfies different conditions when the value ranges of x are different, we have obtained SLLN for function of Markov chain in the environment of single infinite Markovian systems. In addition, the paper studies the strong convergence of the weighted sums of function for finite state Markov Chains in single infinitely Markovian environments. Although the similar conclusions have been carried out, the difference results performed by previous scholars are that we give weaker different sufficient conditions of the strong convergence of weighted sums compared with the previous conclusions.

scholarly journals A central limit property under a modified Ehrenfest urn design

2006 ◽  
Vol 43 (02) ◽  
pp. 409-420 ◽  
Author(s):  
Yung-Pin Chen
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Central Limit ◽  
Renewal Theory ◽  
Urn Model ◽  
Kolmogorov Type ◽  
Maximal Inequalities ◽  
Limit Property ◽  
Minimum Number ◽  
The Difference

We consider a stochastic process in a modified Ehrenfest urn model. The modification prescribes there to be a minimum number of balls in each urn, and the process records the differences between treatment assignments under a sampling scheme implemented with this modified Ehrenfest urn model. In contrast to the result that the difference process forms a Markov chain and converges to a stationary distribution under the Ehrenfest urn model, the corresponding process under this modified Ehrenfest urn design satisfies the central limit property. We prove this asymptotic normality property using a central limit theorem for dependent random variables, renewal theory, and two Kolmogorov-type maximal inequalities.

scholarly journals A Confront between Amati and Combo Correlations at Intermediate and Early Redshifts

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1118
Author(s):  
Marco Muccino
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Predictive Power ◽  
Energy Equation ◽  
Gamma Ray ◽  
Zero Curvature ◽  
Level 3 ◽  
The Difference ◽  
Best Fit ◽  
The Universe

I consider two gamma-ray burst (GRB) correlations: Amati and Combo. After calibrating them in a cosmology-independent way by employing Beziér polynomials to approximate the Observational Hubble Dataset (OHD), I perform Markov Chain Monte Carlo (MCMC) simulations within the Λ CDM and the wCDM models. The results from the Amati GRB dataset do not agree with the standard Λ CDM model at a confidence level ≥ 3 – σ . For the Combo correlation, all MCMC simulations give best-fit parameters which are consistent within 1– σ with the Λ CDM model. Pending the clarification of whether the diversity of these results is statistical, due to the difference in the dataset sizes, or astrophysical, implying the search for the most suited correlation for cosmological analyses, future investigations require larger datasets to increase the predictive power of both correlations and enable more refined analyses on the possible non-zero curvature of the Universe and the dark energy equation of state and evolution.

scholarly journals A central limit property under a modified Ehrenfest urn design

2006 ◽  
Vol 43 (2) ◽  
pp. 409-420 ◽  
Author(s):  
Yung-Pin Chen
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Central Limit ◽  
Renewal Theory ◽  
Urn Model ◽  
Kolmogorov Type ◽  
Maximal Inequalities ◽  
Limit Property ◽  
Minimum Number ◽  
The Difference

We consider a stochastic process in a modified Ehrenfest urn model. The modification prescribes there to be a minimum number of balls in each urn, and the process records the differences between treatment assignments under a sampling scheme implemented with this modified Ehrenfest urn model. In contrast to the result that the difference process forms a Markov chain and converges to a stationary distribution under the Ehrenfest urn model, the corresponding process under this modified Ehrenfest urn design satisfies the central limit property. We prove this asymptotic normality property using a central limit theorem for dependent random variables, renewal theory, and two Kolmogorov-type maximal inequalities.

scholarly journals Phase Enlargement of Semi-Markov Systems without Determining Stationary Distribution of Embedded Markov Chain

2020 ◽  
Vol 19 (3) ◽  
pp. 539-563
Author(s):  
Vadim Kopp ◽  
Mikhail  Zamoryonov ◽  
Nikita Chalenkov ◽  
Ivan Skatkov
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Stationary Distribution ◽  
Phase State ◽  
Random Variables ◽  
Embedded Markov Chain ◽  
Discrete State ◽  
Markov Systems ◽  
The Difference

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space.  Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

scholarly journals Exact Computation of Pearson Statistics Distribution and Some Experimental Results

2016 ◽  
Vol 37 (1) ◽  
Author(s):  
Marina V. Filina ◽  
Andrew M. Zubkov
Keyword(s):  
Markov Chain ◽  
Experimental Results ◽  
Computational Experiments ◽  
Approximate Computation ◽  
Exact Computation ◽  
Chi Square ◽  
The Difference

A Markov chain based algorithms for exact and approximate computation of Pearson statistics distribution for multinomial scheme are described. Results of computational experiments reveal some new properties of the difference between this distribution and corresponding chi-square distribution.

Some Results on an Optimal Maintenance Policy for a Markovian Deteriorating System Subject to Random Shocks

2017 ◽  
Vol 24 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Nobuyuki Tamura
Keyword(s):  
Markov Chain ◽  
Transition Probability ◽  
Maintenance Policy ◽  
Discrete State ◽  
Discounted Cost ◽  
Special Cases ◽  
Optimal Maintenance ◽  
The Difference ◽  
Random Shocks ◽  
Optimal Maintenance Policy

This paper considers a system whose deterioration is modeled as a discrete-time and discrete-state Markov chain and is subject to randomly occurring shocks. The system is more likely to deteriorate after the occurrence of shocks. At each time epoch, we can select from one of three possible actions: operate, repair, or replace. The difference between repair and replace is whether the influence of shocks is removed or not. While repair does not change the number of shocks that the system has suffered, replacement can return it to zero. We derive an expected discounted cost for an unbounded horizon and show that a generalized control limit policy holds under certain assumptions on the costs and the transition probability. Several structural properties of the optimal maintenance policy are also investigated under different assumptions for the occurrence of shocks. These results are useful for numerically determining the optimal maintenance policy. Finally, we consider some special cases by imposing constraints on the costs and the probabilities.

scholarly journals APLIKASI RANTAI MARKOV PADA PENENTUAN HARI BERSALJU DI BEBERAPA KOTA AMERIKA SERIKAT

2020 ◽  
Vol 2 (2) ◽  
pp. 28
Author(s):  
Rofiroh Rofiroh Rofiroh
Keyword(s):  
United States ◽  
New York ◽  
Stochastic Process ◽  
Markov Chain ◽  
Transition Matrix ◽  
Transition Probability ◽  
The United States ◽  
Observation Station ◽  
The Difference ◽  
Step Transition

ABSTRACT This research models on the stochastic process. The method used is the Marchov chain method with the stochastic process where the forthcoming condition   will only be influenced by the closest preceding condition . This method was applied to the observational data snaw day for the Markov chain at eight observation stations in the United States, namely the New York, Sedro Wooley, Glendivem Willow City, Del Norte, Medford, Charlestone, and Blue Hill. The purpose of this study is to determine the convergence direction of the  step transition probability and the probability distribution of the Markov chain in three conditions. According to the results of data processing using matlab software, diagonal matrices, and spectral theorems, similar results were obtained on the convergence of the transition matrix of each observation station which was influenced by the difference in probability changes of two conditions.  Keywords: Marchov Chain, Snaw Day, Transition Matrix ABSTRAK Penelitian ini melakukan pemodelan pada proses stokastik. Metode penelitian yang digunakan adalah metode rantai markov dengan proses stokastik, keadaan yang akan datang  hanya akan  dipengaruhi keadaan terdekat sebelumnya . Metode ini diterapkan pada data pengamatan  hari bersalju untuk rantai markov di delapan stasiun pengamatan yang ada di Amerika Serikat, yaitu stasiun pengamatan New York, Sedro Wooley, Glendive, Willow City, Del Norte, Medford, Charleston, dan Blue Hill. Tujuan penelitian ini adalah untuk mengetahui arah kekonvergenan peluang transisi dan menentukan distribusi peluang rantai markov n langkah dengan tiga keadaan. Berdasarkan hasil pengolahan data dengan menggunakan software matlab, matriks diagonal, teorema spektral didapatkan hasil yang sama untuk kekonvergenan matriks transisi dari masing-masing stasiun pengamatan dipengaruhi oleh selisih perubahan peluang dua keadaan. Kata kunci: Rantai Markov, Hari Bersalju, Matriks Transisi

Uncertainty Quantification of Simplified Viscoelastic Continuum Damage Fatigue Model using the Bayesian Inference-Based Markov Chain Monte Carlo Method

2020 ◽  
Vol 2674 (4) ◽  
pp. 247-260 ◽  
Author(s):  
Jing Ding ◽  
Yizhuang David Wang ◽  
Saqib Gulzar ◽  
Youngsoo Richard Kim ◽  
B. Shane Underwood
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Monte Carlo Method ◽  
Asphalt Concrete ◽  
Dynamic Modulus ◽  
Cyclic Fatigue ◽  
Continuum Damage ◽  
Viscoelastic Continuum Damage ◽  
Damage Characterization ◽  
The Difference

The simplified viscoelastic continuum damage model (S-VECD) has been widely accepted as a computationally efficient and a rigorous mechanistic model to predict the fatigue resistance of asphalt concrete. It operates in a deterministic framework, but in actual practice, there are multiple sources of uncertainty such as specimen preparation errors and measurement errors which need to be probabilistically characterized. In this study, a Bayesian inference-based Markov Chain Monte Carlo method is used to quantify the uncertainty in the S-VECD model. The dynamic modulus and cyclic fatigue test data from 32 specimens are used for parameter estimation and predictive envelope calculation of the dynamic modulus, damage characterization and failure criterion model. These parameter distributions are then propagated to quantify the uncertainty in fatigue prediction. The predictive envelope for each model is further used to analyze the decrease in variance with the increase in the number of replicates. Finally, the proposed methodology is implemented to compare three asphalt concrete mixtures from standard testing. The major findings of this study are: (1) the parameters in the dynamic modulus and damage characterization model have relatively strong correlation which indicates the necessity of Bayesian techniques; (2) the uncertainty of the damage characteristic curve for a single specimen propagated from parameter uncertainties of the dynamic modulus model is negligible compared to the difference in the replicates; (3) four replicates of the cyclic fatigue test are recommended considering the balance between the uncertainty of fatigue prediction and the testing efficiency; and (4) more replicates are needed to confidently detect the difference between different mixtures if their fatigue performance is close.

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