Related papers: Factorial moments of point processes
We derive a simple expression for the $r^{th}$ factorial moment $\mu_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically,…
We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationary spatio-temporal point processes. The statistic is defined through the n-point correlation functions of the point process and it generalises the…
Denoting by $P_N(A,\theta)=\det(I-Ae^{-i\theta})$ the characteristic polynomial on the unit circle in the complex plane of an $N\times N$ random unitary matrix $A$, we calculate the $k$th moment, defined with respect to an average over…
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the…
Affine point processes are a class of simple point processes with self- and mutually-exciting properties, and they have found useful applications in several areas. In this paper, we obtain large-time asymptotic expansions in large…
We present an analytic method for computing the moments of a sum of independent and identically distributed random variables. The limiting behavior of these sums is very important to statistical theory, and the moment expressions that we…
Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point…
The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and…
In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem.…
We show that the hitting times for points of real $\alpha-$stable L\'evy processes ($1<\alpha\le 2$) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the…
Using the results of X. Fernique on the compactness of distributions of cadlag random functions, we derive some cadlaguity moment estimates for stochastic processes with jumps.
In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here the related special numbers are Stirling numbers of the…
We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having different but related parameters. We give examples relating…
Fractional Poisson processes, a rapidly growing area of non-Markovian stochastic processes, are useful in statistics to describe data from counting processes when waiting times are not exponentially distributed. We show that the fractional…
Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the…
A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of…
The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph $G$. We show…
We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing…
A multiplicative identity in law connecting the hitting times of completely asymmetric $\alpha-$stable L\'evy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute…
We give general sufficient conditions to prove the convergence of marked point processes that keep record of the occurrence of rare events and of their impact for non-autonomous dynamical systems. We apply the results to sequential…