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}.

Asymptotic distributions of pontograms

1987 ◽  
Vol 101 (1) ◽  
pp. 131-139 ◽  
Author(s):  
Miklós Csörgő ◽  
Lajos Horváth
Keyword(s):  
Poisson Process ◽  
Covariance Function ◽  
Weighted Approximation ◽  
Asymptotic Distributions ◽  
Uhlenbeck Process ◽  
Intensity Parameter ◽  
Ornstein Uhlenbeck Process ◽  
Image Position

Let{N(x), x ≥ 0} be a Poisson process with intensity parameter λ > 0, and introduceWhen looking for the changepoint in the Land's End data, Kendall and Kendall [7] proved for all 0 < ε1 < 1 − ε2 < 1 thatwhere {V(s), − ∞ < s < ∞} is an Ornstein–Uhlenbeck process with covariance function exp (−|t − s|). D. G. Kendall has posed the problem of replacing ε1 and ε2 by zero or by sequences εi(n) → 0 (n → ∞) (i = 1, 2), in (1·1). In this paper we study the latter problem and also its L2 version. The proofs will be based on the following weighted approximation of Zn.

Some recent developments concerning asymptotic distributions of pontograms

1990 ◽  
Vol 108 (3) ◽  
pp. 559-567 ◽  
Author(s):  
Vera R. Eastwood
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Asymptotic Distributions ◽  
Testing Device ◽  
Analytical Problem ◽  
Intensity Parameter ◽  
Straight Line ◽  
Data Problem ◽  
Recent Developments ◽  
Image Position

The data analytical problem of testing whether an empirical set of n given points in the plane could be considered to contain too many straight line configurations in a situation where the generating mechanism of the n points is unknown, was recently reintroduced to the literature by D. G. Kendall and W. S. Kendall[7]. Taking the Land's end data problem (cf. also Broadbent [1] and further references given there) as the anchor point and motivation for their discussion, D. G. and W. S. Kendall developed a new testing device, called the pontogram. The pontogram is a one- parameter stochastic process defined pointwise bywhere N denotes a Poisson process in [0, 1] with N(0) = 0 and unknown intensity parameter μ > 0 and where R = N(1) gives the number of Poisson events in the entire [0, 1] section.

Approximation of sums by compound Poisson distributions with respect to stop-loss distances

1990 ◽  
Vol 22 (2) ◽  
pp. 350-374 ◽  
Author(s):  
S. T. Rachev ◽  
L. Rüschendorf
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Stop Loss ◽  
Scale Transformations

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 &lt; λ &lt; 1, is an example of such a process.

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.

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.

scholarly journals Statistical Methodology for Large Claims

1977 ◽  
Vol 9 (1-2) ◽  
pp. 1-9 ◽  
Author(s):  
J. Tiago de Oliveira
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Critical Level ◽  
Compound Poisson Process ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Jump Function ◽  
Image Position ◽  
Asymptotic Forms ◽  
Independent And Identically Distributed

The question of large claims in insurance is, evidently, a very important one, chiefly if we consider it in relation with reinsurance. To a statistician it seems that it can be approached, essentially, in two different ways.The first one can be the study of overpassing of a large bound, considered to be a critical one. If N(t) is the Poisson process of events (claims) of intensity v, each claim having amounts Yi, independent and identically distributed with distribution function F(x), the compound Poisson processwhere a denotes the critical level, can describe the behaviour of some problems connected with the overpassing of the critical level. For instance, if h(Y, a) = H(Y − a), where H(x) denotes the Heavside jump function (H(x) = o if x < o, H(x) = 1 if x ≥ o), M(t) is then the number of claims overpassing a; if h(Y, a) = Y H(Y − a), M(t) denotes the total amount of claims exceeding the critical level; if h(Y, a) = (Y − a) H(Y − a), M(t) denotes the total claims reinsured for some reinsurance policy, etc.Taking the year as unit of time, the random variables M(1), M(2) − M(1), … are evidently independent and identically distributed; its distribution function is easy to obtain through the computation of the characteristic function of M(1). For details see Parzen (1964) and the papers on The ASTIN Bulletin on compound processes; for the use of distribution functions F(x), it seems that the ones developed recently by Pickands III (1975) can be useful, as they are, in some way, pre-asymptotic forms associated with tails, leading easily to the asymptotic distributions of extremes.

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.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

scholarly journals On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

2009 ◽  
Vol 46 (3) ◽  
pp. 721-731 ◽  
Author(s):  
Shibin Zhang ◽  
Xinsheng Zhang
Keyword(s):  
Stable Distribution ◽  
Stochastic Integral ◽  
Random Variables ◽  
Transition Density ◽  
Random Variable ◽  
Independent Random Variables ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Tempered Stable Distribution ◽  
Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

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