Properties of the stochastic ordering for discrete distributions and their applications to the renewal sequence generated by a nonhomogeneous Markov chain

2019 ◽  
Vol 97 ◽  
pp. 33-43
Author(s):  
V. V. Golomozyĭ
Keyword(s):  
Markov Chain ◽  
Stochastic Ordering ◽  
Discrete Distributions ◽  
Nonhomogeneous Markov Chain ◽  
Renewal Sequence

scholarly journals Stochastic ordering of classical discrete distributions

2010 ◽  
Vol 42 (2) ◽  
pp. 392-410 ◽  
Author(s):  
Achim Klenke ◽  
Lutz Mattner
Keyword(s):  
Markov Chain ◽  
Negative Binomial ◽  
Stochastic Ordering ◽  
Discrete Distributions ◽  
The Other ◽  
Size Parameter ◽  
Likelihood Ratios ◽  
Case Methods ◽  
Bernoulli Convolutions ◽  
Lévy Measures

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k*})/Q({k*}) ≥ 1 ≥ limk→k*P({k})/Q({k}), with k* and k* denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k*})/Q({k*}) for finite k*. This includes in particular all pairs where P and Q are both binomial (bn1,p1 ≤stbn2,p2 if and only if n1 ≤ n2 and (1 - p1)n1 ≥ (1 - p2)n2, or p1 = 0), both negative binomial (b−r1,p1 ≤stb−r2,p2 if and only if p1 ≥ p2 and p1r1 ≥ p2r2), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

scholarly journals Stochastic ordering of classical discrete distributions

2010 ◽  
Vol 42 (02) ◽  
pp. 392-410 ◽  
Author(s):  
Achim Klenke ◽  
Lutz Mattner
Keyword(s):  
Markov Chain ◽  
Negative Binomial ◽  
Stochastic Ordering ◽  
Discrete Distributions ◽  
The Other ◽  
Size Parameter ◽  
Likelihood Ratios ◽  
Case Methods ◽  
Bernoulli Convolutions ◽  
Lévy Measures

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤st Q can be characterized by their extreme tail ordering equivalent to P({k *})/Q({k *}) ≥ 1 ≥ lim k→k * P({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k *})/Q({k *}) for finite k *. This includes in particular all pairs where P and Q are both binomial (b n 1,p 1 ≤st b n 2,p 2 if and only if n 1 ≤ n 2 and (1 - p 1) n 1 ≥ (1 - p 2) n 2 , or p 1 = 0), both negative binomial (b − r 1,p 1 ≤st b − r 2,p 2 if and only if p 1 ≥ p 2 and p 1 r 1 ≥ p 2 r 2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

Modeling the Genetic Algorithm by a Nonhomogeneous Markov Chain: Weak and Strong Ergodicity

2013 ◽  
Vol 57 (1) ◽  
pp. 144-151 ◽  
Author(s):  
V. S. M. Campos ◽  
A. G. C. Pereira ◽  
J. A. Rojas Cruz
Keyword(s):  
Genetic Algorithm ◽  
Markov Chain ◽  
Strong Ergodicity ◽  
Nonhomogeneous Markov Chain

Discrete distributions related to success runs of lengthk in aulti-state markov chain

1997 ◽  
Vol 26 (4) ◽  
pp. 1031-1049 ◽  
Author(s):  
Nikolai Kolev ◽  
Leda Minkova
Keyword(s):  
Markov Chain ◽  
Discrete Distributions ◽  
Success Runs

Monotonicity results for MR/GI/1 queues

1997 ◽  
Vol 34 (02) ◽  
pp. 514-524 ◽  
Author(s):  
Nicole Bäuerle
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Transition Matrix ◽  
Stochastic Ordering ◽  
Arrival Process ◽  
Stationary Waiting Time ◽  
Monotonicity Results

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

Testing Statistical Hypotheses on Stochastic Ordering of Discrete Distributions

2011 ◽  
Vol 30 (3) ◽  
pp. 249-260
Author(s):  
Myron N. Chang
Keyword(s):  
Stochastic Ordering ◽  
Discrete Distributions ◽  
Statistical Hypotheses

On a class of renewal functions

1965 ◽  
Vol 61 (2) ◽  
pp. 519-526 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Recurrent Events ◽  
Discrete Distributions ◽  
Recurrent Event ◽  
Frequency Function ◽  
Renewal Processes ◽  
Density Functions ◽  
Renewal Sequence ◽  
Analogous Property ◽  
Image Position ◽  
Joint Occurrence

Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described by renewal sequences {un} which are generated by discrete distributions , according to the equationAny two renewal sequences {u′n}, {u″n} define another renewal sequence {un} by means of their term-by-term product {un} = {u′nu″n}, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency function f(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functions h1(x) and h2(x), the function h(x) = h1(x) h2(x) is a renewal density function. A little intuitive reflexion shows that while h(x) dx has a probability density interpretation, this is not in general true of h1(x) h2(x) dx. It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densities h(x) for which αh(x) is a renewal density for all finite positive α and not merely α in 0 < α ≤ A < ∞. In turn this characterization enables us to define a class of renewal densities for which h1(x) and imply that .

scholarly journals Limit theorems for asymptotic circular mth-order Markov chains indexed by an m-rooted homogeneous tree

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1817-1832
Author(s):  
Huilin Huang ◽  
Weiguo Yang
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Nonhomogeneous Markov Chain ◽  
Asymptotic Equipartition Property ◽  
Large Numbers ◽  
Definition Of ◽  
Rooted Homogeneous Tree

In this paper, we give the definition of an asymptotic circularmth-order Markov chain indexed by an m rooted homogeneous tree. By applying the limit property for a sequence of multi-variables functions of a nonhomogeneous Markov chain indexed by such tree, we estabish the strong law of large numbers and the asymptotic equipartition property (AEP) for asymptotic circular mth-order finite Markov chains indexed by this homogeneous tree. As a corollary, we can obtain the strong law of large numbers and AEP about the mth-order finite nonhomogeneous Markov chain indexed by the m rooted homogeneous tree.

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