Questions tagged [point-processes]

Question 1

Suppose you have a Poisson Point Process $\{X_k\}$ with intensity $1$ on $\mathbb{R}$, and we truncate it on $[0,L]$. For a positive integer $n$, I would like to compute or estimate \begin{equation*} \...

Question 2

Lemma 2.5(Property of Voronoi tessellations) of the book "On Einstein's viscosity formula" states (here we restate a simplified version): Given an underlying probability space $(\Omega,\...

Question 3

Suppose we have a Toeplitz matrix $K$ which is $N\times N$. I want to simulate the determinantal point process associated with this matrix. Let's just start with something simple, like $N=2$. So $$ K=\...

Question 4

While studying Hawkes processes, I have come across a textbook (Laub, Lee, and Taimre) and many articles that mention (but do not prove) that, if $N$ is the counting process of a Hawkes process, $\...

Question 5

The conditions of Campbell's Theorem says that $\eta$ is a point process with intensity measure $\lambda$, (i.e. $E(\eta(B))=\lambda(B)$ for all measurable $B$) then for a function $f$ such that $f\...

Question 6

I am currently learning about determinantal point processes, mainly via this survey by Kulesza and Taskar. The likelihood kernel $L$, a $N\times N$ symmetric positive semidefinite matrix, of such a ...

Question 7

Let $\{X_n\}_{n=1}^\infty$ be a real-valued stationary stochastic process, and let $\{W_n\}_{n=1}^\infty$ be a binary-valued stochastic process, where $W_n \in \{0, 1\}$. We call $W_n$ the event ...

Question 8

Suppose we have some point process $N=N_{\theta_0}$ on the real line, driven by a conditional intensity $\lambda_{\theta_0}$ dependent on some finite-dimensional parameter $\theta_0\in\Theta\subset\...

Question 9

In the spatial point process literature, it is customary to specify a point process by assigning its density with respect to a Poisson point process. For instance, an Hardcore point process $\mathbf X$...

Question 10

I am seeking to use the following theorem by Kallenberg (1975) (the version quoted below is from Brown, 1979) Theorem. Let $\{N_n\}$ be a sequence of point processes and $\{p_n\}$ a sequence in $(0,1]...

Question 11

Consider a continuous time point process $\eta(t)$ representing the number of children in the interval $[0,t]$. Then $\eta(\infty)$ gives the total number of children of an individual. We define $\mu(...

Question 12

Let $(Z_t)_{t\geq0}$ be a homogeneous Poisson point process of intensity $\lambda>0$, such that to each event at time $t$, there is some random location $x\in[0,1]$ attached. More specifically, ...

Question 13

Consider a continuous time point process $\eta(t)$ representing the number of points in the interval $[0, t]$. Let $\eta(\infty)$ be distributed as the total number of children of a particle. Define $\...

Question 14

Let $(\Omega, F, \mathbb{P})$ be a probability space and let $(S, \Sigma)$ be a measurable space. Let $S^{\infty}$ be the collection of countable subsets of $S$. Assume that $\left\{x\right\} \in \...

Question 15

I am struggling with Theorem 3.5 from the paper of Davis and Mikosh "THE SAMPLE AUTOCORRELATIONS OF HEAVY-TAILED PROCESSES WITH APPLICATIONS TO ARCH". This paper is available, for example, ...

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Questions tagged [point-processes]

Lengths of Segments Formed by Binomial Point Process

Prove $\mathbb{E}[\zeta] = \mathbb{E}\Biggl[\, \sum_{n}\, \frac{\mathbf{1}_{\{0\in I_n\}}}{\lvert I_n\rvert} \int_{V_n} \zeta \Biggr]$.

Simulating discrete determinantal point process

Problem with Markovian Hawkes Process / Intensity Definition?

Failure of Campbell's Theorem in Point Processes

Eigendecomposition of a Gram Matrix with Specified Columns

Convergence of subsequences of a stationary stochastic process

Difference between compensator of point process under real parameter an its MLE estimator

Janossy densities and the relation with density with respect to a Poisson point process

Interpretation of $p_n N_n$ in Kallenberg's theorem on thinning of point processes?

Existence of Malthusian Parameter (general branching process)

CLT for spatiotemporal processes having a white noise limit

Existence of Malthusian parameter

Measurability condition of point processes

Point process representation of the limit of ACF

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