Multivariate extremal processes generated by independent non-identically distributed random variables

1975 ◽  
Vol 12 (03) ◽  
pp. 477-487 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Extremal Processes

Letbe thekth largest amongXn1, …,Xn[nt], whereXni= (Xi– an)/bn, {Xi} is a sequence of independent random variables andbn> 0 andanare norming constants. Suppose that for eachconverges in distribution. Then all the finite-dimensional laws ofconverge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.

Multivariate extremal processes generated by independent non-identically distributed random variables

1975 ◽  
Vol 12 (3) ◽  
pp. 477-487 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Extremal Processes

Let be the kth largest among Xn1, …, Xn[nt], where Xni = (Xi – an)/bn, {Xi} is a sequence of independent random variables and bn > 0 and an are norming constants. Suppose that for each converges in distribution. Then all the finite-dimensional laws of converge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (01) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X 1, X 2, … are non-constant mutually independent random variables with X 1,X 2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X 1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

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.

Asymptotic distributions of weighted compound Poisson bridges

1992 ◽  
Vol 112 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Barbara Szyszkowicz
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Independent Random Variables ◽  
Intensity Parameter ◽  
Compound Poisson ◽  
Image Position

Let S(N(t)) be defined bywhere {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.

scholarly journals The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables

2010 ◽  
Vol 47 (04) ◽  
pp. 908-922 ◽  
Author(s):  
Yiqing Chen ◽  
Anyue Chen ◽  
Kai W. Ng
Keyword(s):  
Law Of Large Numbers ◽  
Large Deviation ◽  
Random Variables ◽  
Renewal Theory ◽  
Independent Random Variables ◽  
Strong Law ◽  
Finite Dimensional ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
New Inequalities

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (1) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

Estimation of a nonlinear functional from the probability density when optimizing nonparametric decision functions

2021 ◽  
pp. 14-20
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Probability Density ◽  
Decision Rules ◽  
Random Variables ◽  
Random Variable ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Probability Density Estimation ◽  
Nonlinear Functional ◽  
Approximation Errors

A method for estimating the nonlinear functional of the probability density of a two-dimensional random variable is proposed. It is relevant when implementing procedures for fast bandwidths selection in the problem of optimization of kernel probability density estimates. The solution of this problem allows to significantly improve the computational efficiency of nonparametric decision rules. The basis of the proposed approach is the analysis of the formula for the optimal bandwidth of the kernel probability density estimation. In this case, the bandwidth of kernel functions is represented as the product of an indeterminate parameter and the average square deviations of random variables. The main component of an undefined parameter is a nonlinear functional of the probability density. The considered functional is determined by the type of probability density and does not depend on the density parameters. For a family of two-dimensional lognormal laws of distribution of independent random variables, the approximation errors of the considered nonlinear functional from the probability density are determined. The possibility of applying the proposed methodology when evaluating nonlinear functionals of probability densities that differ from the lognormal distribution laws is investigated. An analysis is made of the effect of the resulting approximation errors on the root-mean-square criteria for restoring a non-parametric estimate of the probability density of a two-dimensional random variable.

The two-dimensional Poisson process and extremal processes

1971 ◽  
Vol 8 (4) ◽  
pp. 745-756 ◽  
Author(s):  
James Pickands
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Order Statistics ◽  
Sample Space ◽  
Two Dimensional ◽  
Record Times ◽  
Extreme Order Statistics ◽  
Extremal Processes

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

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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