Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (03) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (3) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

Optimal Advertizing Policy for Selling a Single Asset

1991 ◽  
Vol 5 (1) ◽  
pp. 89-100 ◽  
Author(s):  
David Assaf ◽  
Benny Levikson
Keyword(s):  
Poisson Process ◽  
Nonhomogeneous Poisson Process ◽  
Independent And Identically Distributed

Suppose we have a single asset that we would like to sell. As time goes by, independent and identically distributed offers with a common known distribution F are given to us. At any given moment, we may either accept the current offer or reject it, thereby losing it forever. The rate at which offers arrive follows a nonhomogeneous Poisson process whose instantaneous intensity is under our control, using advertizing in a manner to be described. Our objective is, roughly, that of maximizing the total discounted expected reward composed of the offer we decide to accept, minus the total advertizing costs.

Some results on a traffic model of Rényi

1969 ◽  
Vol 6 (02) ◽  
pp. 293-300
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Constant Velocity ◽  
Random Variables ◽  
Traffic Model ◽  
Homogeneous Poisson Process ◽  
Image Position

In [5] Renyi considers the following traffic model: Vehicles enter a highway at times 〈Ti , i = 1, 2, … 〉, forming a homogeneous Poisson process of intensity λ. The vehicle entering at time Ti will choose a velocity Vi and will travel at that constant velocity. The random variables 〈Vi , i = 1, 2, …〉 are independently and identically distributed (i.i.d.) and independent of 〈Ti 〉 with c.d.f. F satisfying All vehicles travel in the same direction.

On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

1986 ◽  
Vol 23 (01) ◽  
pp. 221-226 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Arbitrary Point ◽  
Limiting Distribution ◽  
Poisson Processes ◽  
Random Point ◽  
Independent Random Variables ◽  
Nearest Neighbour ◽  
Homogeneous Poisson Process

In a homogeneous Poisson process in R d , consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (02) ◽  
pp. 453-458 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Random Variables ◽  
Poisson Processes ◽  
Invariance Property ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Random Functions

In this paper we shall investigate point processes generated by random variables of the form 〈gi (Ti ]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti 〉.

Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

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