Peacocks and the Zeta distributions

2021 ◽  
Author(s):  
El ghazi Imad
Keyword(s):  
Stochastic Process ◽  
Random Variable ◽  
Short Paper ◽  
Convex Order ◽  
Increasing Process

Abstract We prove in this short paper that the stochastic process defined by: $$Y_{t} := \frac{X_{t+1}}{\mathbb{E}\left[ X_{t+1}\right]},\; t\geq a > 1,$$ is an increasing process for the convex order,where Χt a random variable taking values in N with probability P(Χt = n) = n-t/(𝛇(t)) and 𝛇(t) = +∞∑k=1(1/kt), ∀t > 1.

A thermal energy storage process with controlled input

1982 ◽  
Vol 14 (02) ◽  
pp. 257-271 ◽  
Author(s):  
D. J. Daley ◽  
J. Haslett
Keyword(s):  
Stochastic Process ◽  
Energy Storage ◽  
Thermal Energy ◽  
Thermal Energy Storage ◽  
Queueing Theory ◽  
Random Variable ◽  
Stationary Sequence ◽  
Storage Process ◽  
Storage Model ◽  
Special Cases

The stochastic process {Xn } satisfying Xn +1 = max{Yn +1 + αβ Xn , βXn } where {Yn } is a stationary sequence of non-negative random variables and , 0<β <1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study of μ = EX where the random variable X has the stationary distribution for {Xn }. Even for special cases such as i.i.d. Yn or α = 0, little explicit information appears to be available on the distribution of X or μ . Accordingly, bounding techniques that have been exploited in queueing theory are used to study μ . The various bounds are illustrated numerically in a range of special cases.

A characterization of stable processes

1969 ◽  
Vol 6 (02) ◽  
pp. 409-418 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Interior Point ◽  
Random Variable ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

On the normal convergence of a class of simple batch epidemics

1981 ◽  
Vol 18 (01) ◽  
pp. 31-41
Author(s):  
Naftali A. Langberg
Keyword(s):  
Stochastic Process ◽  
Mild Condition ◽  
Random Variable ◽  
Normal Random Variable ◽  
Contagious Disease ◽  
Normal Convergence

A group of n susceptible individuals exposed to a contagious disease is considered. It is assumed that at each instant in time one or more susceptible individuals can contract the disease. The progress of this epidemic is modeled by a stochastic process Xn (t), t in [0,∞) representing the number of infective individuals at time t. It is shown that Xn (t), with the suitable standardization and under a mild condition, converges in distribution as n → ∞to a normal random variable for all t in (0, t 0), where t 0 is an identifiable number.

scholarly journals Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
M.-C. Casabán ◽  
J.-C. Cortés ◽  
J.-V. Romero ◽  
M.-D. Roselló
Keyword(s):  
Stochastic Process ◽  
Probability Density Function ◽  
Differential Equations ◽  
Probability Density ◽  
Density Function ◽  
Initial Value Problems ◽  
Random Variable ◽  
Initial Value ◽  
Variable Transformation ◽  
Comprehensive Study

Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

Existence and positivity of the limit in processes with a branching structure

1999 ◽  
Vol 36 (1) ◽  
pp. 132-138
Author(s):  
M. P. Quine ◽  
W. Szczotka
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Random Variables ◽  
Random Variable ◽  
Natural Generalization ◽  
Partial Sums ◽  
Urn Model ◽  
Special Case ◽  
Independent And Identically Distributed

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

scholarly journals On the convergence of quadratic variation for compound fractional Poisson processes

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Enrico Scalas ◽  
Noèlia Viles
Keyword(s):  
Stochastic Process ◽  
Random Variable ◽  
Quadratic Variation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
The Relationship ◽  
Compound Renewal Processes

AbstractThe relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

Does Network Theory Contradict Trait Theory?

2012 ◽  
Vol 26 (4) ◽  
pp. 447-448 ◽  
Author(s):  
Rolf Steyer
Keyword(s):  
Stochastic Process ◽  
Network Theory ◽  
Latent Variable ◽  
Random Variable ◽  
Time Points ◽  
Trait Theory ◽  
Colloquial Language

I argue that the trait and network theories of personality are not necessarily contradictory. If appropriately formalized, it may turn out that network theory incorporates traits as part of the theory. I object the opinion that if a trait is a cause of behaviour, then it is necessarily an entity operating in the minds of individuals. Finally, I argue that liking parties can be a label for a random variable (item), a stochastic process (a family of items at different time points) and a latent variable (trait). In our colloquial language, we do not make these distinctions, which leads often to confusions. Copyright © 2012 John Wiley & Sons, Ltd.

scholarly journals On the optimal assignment of customers to parallel servers

1978 ◽  
Vol 15 (02) ◽  
pp. 406-413 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Stochastic Process ◽  
Service Time ◽  
Hazard Rate ◽  
Random Variable ◽  
Queuing System ◽  
Optimal Assignment ◽  
Parallel Servers ◽  
Shortest Queue ◽  
Number Of Customers

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some server's queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customer's service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue.

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