# Questions tagged [point-processes]

This tag is for questions concerning point processes such as poisson point processes or any other point process.

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### Can't understand the proof of the Time-Rescaling theorem.

I was reading the following paper: The time-rescaling theorem and its application to neural spike train data analysis and I have some difficulties understanding the proof of the time-rescaling-theorem....
0answers
16 views

### Why is a completely random measure mixing?

Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite ...
1answer
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### Point Process as a Random Distribution of Indistinguishable Points

I am currently reading Point Processes and Their Statistical Inference (2nd Edition) and had a question about how point processes are defined on page 5. So a point process on $E$ is defined as a ...
0answers
13 views

### Expected value of hawkes process

I’m studying hawkes processes and I find the formulation of the expected value of them in integration. Actually I’m here to ask is there any closed form for expected value and other measures for multi-...
0answers
27 views

### Expectation of Hawkes process with exponential kernel

Let N be a point process adapted to a filtration $\mathcal{F}_{t}$. The left-continuous intensity process is defined as \begin{equation} \begin{split} \lambda(t|\mathcal{F}_{t})&=\lim_{h\...
0answers
13 views

### Non parametric estimation of point process entropy

Let point process $\Phi$ be defined on some bounded set $A$ in the borel $\sigma$-algebra on $\mathbb{R}^d$, and assume that the parametric structure of $\Phi$ is unknown. Let $\{x_1,\ldots,x_N\}$ be ...
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### Realizations from a Poisson point process in abstract spaces

Let $(X, d)$ a complete and separable metric space and $G$ a $\sigma$-finite measure on $(X, \mathcal B(X))$. From Kingman , I know that a Poisson point process $\Pi$ on $(X, d)$ is a random ...
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19 views

### Formal proof distribution of interarrival times of a HPP

Let $\{N(t)\}_{t\ge0}$ be a Poisson counting process with intensity $\lambda>0$ and $\{T_n\}_{n\in\mathbb{N}}$ the associated point process. Then we know that the interarrival times $T_{n+1}-T_n$ ...
1answer
44 views

### Expected measure of a ball in a probability space with a metric

Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms ...
2answers
87 views

### Long run percentage of customers who wait for a bus less than x units of time if customers arrive according to a homogenous Poisson process?

Assume that customers arrive to a bus stop according to a homogenous Poisson process with rate $\alpha$ and that the arrival process of buses is an independent renewal process with interarrival ...
1answer
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1answer
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### Intensity function $\lambda(u)$ of non-stationary MatérnI hard-core point process?

MatérnI description In a MatérnI hard-core process, a stationary PPP $\Phi$ defined at $\mathbb{R}^d$ with intensity $\lambda$ is generated. Then the points are removed if there exists others lying ...
1answer
240 views

### What is the intensity measure of a thinned Poisson point process?

Scenario I have a non-homogeneous Poisson point process (PPP) $X\in\mathbb{R}^2$ with intensity function $\lambda(u)$ that is observed over a bounded region $W$. This PPP is modifyed by a dependent ...
1answer
112 views

### Why this definition of spherical contact distribution function is $1 - N(b(o,r) =0)$ and not $N(b(o,r) =0)$?

I've been doing some reading on spatial Poisson point processes on my own tonight, and right now having a headache or a brainwarp or I don't know what because I don't get this definition on Wikipedia: ...
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178 views

### Hands-On Matlab Resources for Wireless Networks Modeling using Stochastic Geometry

Stochastic Geometry has become a very strong mathematical tool for studying and understanding several aspects of wireless communication and networks. As I write this, I find quite a large number of ...
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60 views

### Average sum of distances of Poisson point process falling in Poisson-Voronoi cells

Exercise Having two homogeneous and independent Poisson point processes $\Phi_3, \Phi_2$ defined in $\mathbb{R}^2$ with intensities $\lambda_3, \lambda_2$, respectively. Having a Voronoi tessellation ...
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39 views

I'm self-taught studying Poisson point processes and I can't understand the proof of existence in the theorem that states that a Poisson Process is uniquely determined given a locally finite measure $\... 0answers 69 views ### What does the weak convergence of stochastic intensity tell us about the point process? Suppose we have a sequence of marked point processes$N_n$on the same filtration space,$(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$with$\mathcal{F}_t$-predictable intensities$\lambda_n(t,k)$. ... 1answer 74 views ### Modelling Poisson “Point” Process and data transmission with Poisson process If a Poisson Point Process (PPP)$\Phi_c$with density$\lambda_c$(points/m$^2$) distributed over 2D plane. These points depict the cellular nodes. Consider every node transmit data on the uplink ... 1answer 42 views ### expectations in poisson point process here,$\Phi_e$is a poisson point process and$\eta_k$a random variable having exponential distribution. I'm having trouble in understanding how this equality holds? 0answers 74 views ### Probability of$N \ge n\$ points for an inhomogeneous poisson point process

I am trying to figure out the probability of at least n points for an inhomogeneous poisson point process defined on the real line. $$P\{N(a,\infty) \ge n \} = ?$$ I'm also not entirely sure if ...