Parametric stochastic convexity and concavity of stochastic processes

1990 ◽  
Vol 42 (3) ◽  
pp. 509-531 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Stochastic Processes ◽  
Stochastic Convexity ◽  
Convexity And Concavity

scholarly journals Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Fangfang Ma ◽  
Waqas Nazeer ◽  
Mamoona Ghafoor
Keyword(s):  
Probability Theory ◽  
Convex Function ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Probabilistic Models ◽  
Random Variable ◽  
Expected Value ◽  
Integral Inequalities ◽  
Stochastic Convexity ◽  
Convex Stochastic Processes

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized p -convex stochastic processes. Some well-known results of generalized p -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p -stochastic convexity.

Regularity of Stochastic Processes: A Theory Based on Directional Convexity

1993 ◽  
Vol 7 (3) ◽  
pp. 343-360 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Stochastic Processes ◽  
Large Class ◽  
Convex Functions ◽  
Queueing Systems ◽  
Single Stage ◽  
Renewal Processes ◽  
Doubly Stochastic ◽  
Directional Convexity ◽  
Stochastic Convexity ◽  
Markov Renewal Processes

We define a notion of regularity ordering among stochastic processes called directionally convex (dcx) ordering and give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Further-more, we show that the class of segmented processes introduced by Chang, Chao, and Pinedo [3] provides a rich set of stochastic processes where the dcx ordering can be commonly encountered. When the input processes to a large class of queueing systems (single stage as well as networks) are dcx ordered, so are the processes associated with these queueing systems. For example, if the input processes to two tandem /M/c1→/M/c2→…→/M/cm queueing systems are dcx ordered, so are the numbers of customers in the systems. The concept of directionally convex functions (Shaked and Shanthikumar [15]) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo, and Shanthikumar [4]) are employed in our analysis.

Stochastic convexity and its applications

1988 ◽  
Vol 20 (02) ◽  
pp. 427-446 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Queueing Theory ◽  
Empirical Distribution ◽  
Branching Processes ◽  
Sample Path ◽  
Arrival Rate ◽  
Stochastic Convexity ◽  
Parametric Statistics ◽  
Service Rates ◽  
Convexity And Concavity ◽  
Probability And Statistics

Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady-state waiting time in a GI/D/c queue, and (as a function of the arrival or service rates) in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case (as a function of the arrival rate) is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff in the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In non-parametric statistics, the stochastic concavity (or convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

Stochastic Convexity and Concavity of Markov Processes

1994 ◽  
Vol 19 (2) ◽  
pp. 477-493 ◽  
Author(s):  
Haijun Li ◽  
Moshe Shaked
Keyword(s):  
Markov Processes ◽  
Stochastic Convexity ◽  
Convexity And Concavity

Stochastic convexity and its applications

1988 ◽  
Vol 20 (2) ◽  
pp. 427-446 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Queueing Theory ◽  
Empirical Distribution ◽  
Branching Processes ◽  
Sample Path ◽  
Arrival Rate ◽  
Stochastic Convexity ◽  
Parametric Statistics ◽  
Service Rates ◽  
Convexity And Concavity ◽  
Probability And Statistics

Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady-state waiting time in a GI/D/c queue, and (as a function of the arrival or service rates) in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case (as a function of the arrival rate) is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff in the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In non-parametric statistics, the stochastic concavity (or convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

scholarly journals Temporal stochastic convexity and concavity

1987 ◽  
Vol 27 ◽  
pp. 1-20 ◽  
Author(s):  
Moshe Shaked ◽  
J.George Shanthikumar
Keyword(s):  
Stochastic Convexity ◽  
Convexity And Concavity

Statistical Analysis of Stochastic Processes in Time

2004 ◽  
Author(s):  
J. K. Lindsey
Keyword(s):  
Statistical Analysis ◽  
Stochastic Processes

Stochastic Processes in Demography and Applications

1992 ◽  
Vol 46 (1) ◽  
pp. 172-173
Author(s):  
S. Mitra
Keyword(s):  
Stochastic Processes
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