Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (3) ◽  
pp. 614-620 ◽  
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

2012 ◽  
Vol 26 (2) ◽  
pp. 221-244 ◽  
Author(s):  
M. L. Chaudhry ◽  
S. K. Samanta ◽  
A. Pacheco
Keyword(s):  
Length Distribution ◽  
Renewal Theory ◽  
Computationally Efficient ◽  
Form Analysis ◽  
Matrix Geometric Method ◽  
Alternative Approach ◽  
The Matrix ◽  
Sojourn Time Distribution ◽  
System Length ◽  
Markov Renewal Theory

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

PROBABILITY DISTRIBUTIONS OF SCATTERED SIGNALS IN A RANDOM MULTILAYER

2007 ◽  
Vol 07 (04) ◽  
pp. R49-R61
Author(s):  
J. -H. KIM
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Turning Point ◽  
Review Paper ◽  
Probability Distributions ◽  
Strong Mixing ◽  
Random Wave ◽  
Mixing Condition ◽  
Asymptotic Formulation ◽  
Strong Mixing Condition

This is a review paper on the study of the randomly scattered signals in a random multilayer based upon a stochastic and asymptotic formulation under strong mixing condition. This formulation generalizes the dominant Ito's formulation. The existence of a turning point of the random wave requires several type stochastic differential equations and the relevant limit theorems. The probability distributions of the randomly scattered signals have been obtained in the form of the Kolmogorov PDEs along the line of Khasminskii's limit theorem. This article demonstrates the step-by-step development of the relevant generators which contain the ultimate information for the probability distributions of the random signals.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

scholarly journals Fluctuation of Matrix Entries and Application to Outliers of Elliptic Matrices

2018 ◽  
Vol 70 (1) ◽  
pp. 3-25
Author(s):  
Florent Benaych-Georges ◽  
Guillaume Cébron ◽  
Jean Rochet
Keyword(s):  
Asymptotic Behavior ◽  
Random Matrix ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Asymptotic Independence ◽  
Single Ring ◽  
The Matrix ◽  
Additive Perturbation

AbstractFor any family of N ⨯ N randommatrices that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type Tr(AkM), where the matrix M is deterministic (such random variables include, for example, the normalized matrix entries of Ak). A consequence is the asymptotic independence of the projection of the matrices Ak onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Formof the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other.These phenomena have already been observed with random matrices from the Single Ring Theorem.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (01) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (01) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L 2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

Limit theorems for the number of occurrences of consecutive k successes in n Markovian trials

1995 ◽  
Vol 32 (03) ◽  
pp. 727-735 ◽  
Author(s):  
Y. H. Wang ◽  
Shuixin Ji
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Limiting Distributions ◽  
Compound Poisson ◽  
Stationary Transition

We present a method of deriving the limiting distributions of the number of occurrences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string of S of exact length k preceded and followed by an F, except the first run which may not be preceded by an F, or the last run which may not be followed by an F. 2. A string of S of length k or more. 3. A string of S of exact length k, where recounting starts immediately after a run occurs. 4. A string of S of exact length k, allowing overlapping runs. It is shown that the limits are convolutions of two or more distributions with one of them being either Poisson or compound Poisson, depending on the type of runs in question. The completely stationary Markov case and the i.i.d. case are also treated.

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376 ◽  
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Sign in / Sign up

Export Citation Format

Share Document