A note on seasonal Markov chains with gamma or gamma-like distributions

1979 ◽  
Vol 16 (1) ◽  
pp. 117-128 ◽  
Author(s):  
E. H. Lloyd ◽  
S. D. Saleem
Keyword(s):  
Markov Chain ◽  
Laplace Transform ◽  
Weighted Sums ◽  
Power Transformation ◽  
Exponential Form ◽  
Continuous State ◽  
Bessel Distribution ◽  
Storage Theory ◽  
Special Case ◽  
Type Of Transition

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt} has a Laplace transform (LT) L(Xt+1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it.A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968).A seasonal version of this process is developed, valid for any number of seasons.Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

A note on seasonal Markov chains with gamma or gamma-like distributions

1979 ◽  
Vol 16 (01) ◽  
pp. 117-128 ◽  
Author(s):  
E. H. Lloyd ◽  
S. D. Saleem
Keyword(s):  
Markov Chain ◽  
Laplace Transform ◽  
Weighted Sums ◽  
Power Transformation ◽  
Exponential Form ◽  
Continuous State ◽  
Bessel Distribution ◽  
Storage Theory ◽  
Special Case ◽  
Type Of Transition

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt } has a Laplace transform (LT) L(Xt +1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it. A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968). A seasonal version of this process is developed, valid for any number of seasons. Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (1) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

Sums and weighted sums of a gamma Markov sequence

1988 ◽  
Vol 25 (01) ◽  
pp. 204-209 ◽  
Author(s):  
Ravindra M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Random Variables ◽  
Laplace Transforms ◽  
Weighted Sums ◽  
Sums Of Random Variables ◽  
Markov Sequence

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (01) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

A STORAGE SYSTEM WITH SPORADIC AND CONTINUOUS CLEARINGS

2007 ◽  
Vol 21 (4) ◽  
pp. 539-549 ◽  
Author(s):  
Wolfgang Stadje
Keyword(s):  
Laplace Transform ◽  
Renewal Process ◽  
Storage System ◽  
Renewal Processes ◽  
Finite Capacity ◽  
Special Case ◽  
Independent And Identically Distributed

We study a cumulative storage system that is totally cleared sporadically at stationary renewal times and whenever a finite-capacity threshold is exceeded. The independent and identically distributed inputs occur at time epochs that also form a stationary renewal process. We determine the distribution of the interoverflow times. Although this distribution is quite intricate when both underlying renewal processes are general, in the special case of Poisson sporadic clearings we obtain a neat formula for its Laplace transform.

The extremal index for a Markov chain

1992 ◽  
Vol 29 (01) ◽  
pp. 37-45 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Extreme Values ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extremal Index ◽  
Scaling Properties ◽  
Continuous State Space ◽  
Continuous State ◽  
Tail Behaviour

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.

Modeling Wind Speed Time Series Using Continuous State Markov Chain

2016 ◽  
Author(s):  
Yu-Dun Li ◽  
Yun-Tao Sun ◽  
Chao Yang ◽  
Xin Wang ◽  
Guo-Hui Zhang
Keyword(s):  
Time Series ◽  
Markov Chain ◽  
Wind Speed ◽  
Continuous State

scholarly journals Markov-modulated single-server queueing systems

1994 ◽  
Vol 31 (A) ◽  
pp. 169-184
Author(s):  
N. U. Prabhu ◽  
L. C Tang
Keyword(s):  
Markov Chain ◽  
Queueing Systems ◽  
Classical Case ◽  
Single Server ◽  
Discrete Time Markov Chain ◽  
Joint Distributions ◽  
Markov Random Walk ◽  
Countable State ◽  
Markov Modulated ◽  
Special Case

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.

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.

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