Questions tagged [point-processes]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
19 views

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....
0
votes
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 ...
0
votes
1answer
25 views

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 ...
0
votes
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-...
2
votes
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\...
0
votes
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 ...
0
votes
0answers
23 views

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 [1], I know that a Poisson point process $\Pi$ on $(X, d)$ is a random ...
0
votes
0answers
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 $T_{n+1}-T_n$ ...
1
vote
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 ...
3
votes
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 ...
1
vote
1answer
60 views

Stochastic integration by parts for random point processes

I'm trying to understand this proof of the following specifing integration by parts. Introduction Let $\Omega=Point_{\mathbb{R}}$ the set of point distributions in $\mathbb{R}^3$ (i.e an element $w \...
0
votes
0answers
11 views

Probability of an event regarding finite point processes

I'm studying the theory of finite point processes on "An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, Second Edition" by Daley and Vere-Jones. I ...
1
vote
0answers
56 views

Meaning of Janossy densities

I'm studying the theory of finite point processes on "An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, Second Edition" by Daley and Vere-Jones. I ...
1
vote
0answers
29 views

Conditional intensity of a generalized Poisson process

Assume that we have a temporal point process on the interval [0, 1], where the number of events $N$ is distributed according to some PMF $p(N = n)$. Conditioned on $N$, the arrivals times of the ...
1
vote
0answers
15 views

Mean value for the number of neighbors of typical cell on a manifold

For Poisson-Voronoi tessellations of $\mathbb R^2$, the expected number of vertices on the boundary of the typical cell is 6. Proofs of this can be found in section 9.3.4 of Stochastic Geometry and ...
0
votes
1answer
56 views

Palm formula for Poisson processes

I'm reading some lecture notes and the following gets stated without proof If $N \sim Poisson(\lambda)$ and $f$ and $G$ are functions of $x \in S$ and of $(x, N)$ respectively, then $$ \mathbb E \int ...
0
votes
1answer
109 views

Difference between stationary and homogeneous point process

I do not understand the difference between a stationary and a homogeneous point process. The definitions I found are as follows: A process is stationary if the entire configuration of the process is ...
0
votes
0answers
6 views

Considering dependency between two data point processes

How can we consider dependency between two point process streams? Does multi-dimensional hawkes process an approach?
0
votes
0answers
7 views

Numerical example for considering dependency in point processes

I am searching for a way to consider dependency between two data set witch fitted on two point process streams. Existing temporal dependency between data is the reason that I couldn't use correlation ...
3
votes
1answer
75 views

Compute $ P\{ \text{Harry commits himself in } \left[ 0, t \right]\}.$

Due to the stress of coping with business, Harry begins to experience migraine headaches of random severities. The times when headaches occur follow a Poisson processes of rate $\lambda$. Headache ...
0
votes
0answers
43 views

Stochastically dominated Poisson Process

How can I prove, that a Poisson Point Process $\mathcal{P}_n$ on $\mathbb{R}^d$ of the region $B_{r_n}(x)$, $x\in\mathbb{R}^d$ with intensity $nf$ is stochastically dominated by a Poisson random ...
3
votes
1answer
54 views

Laplace functional of cluster process

Consider the simple cluster process: $$\sum_n \xi_n \epsilon_{X_n}$$ where $\{X_n\}$ are Poisson points independent of the iid non-negative integer sequence $\{\xi_n\}$. How do I find the Laplace ...
5
votes
0answers
53 views

Definition of Random Measures

Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from? I would like to have a motivation ...
1
vote
0answers
52 views

Why are $T_2/T_1, …, T_k/T_{k-1}, T_k$ independent conditional on $N((0,t])=k$, where $N((0,t]) = \sum_i \epsilon_{T_K}((0,t])$ is a point process?

Why are $T_2/T_1, ..., T_k/T_{k-1}, T_k$ independent conditional on $N(t)=k$, where $N(t) = N((0,t]) = \sum_i \epsilon_{T_K}((0,t])$ is a point process, $0,<T_1<T_2<...$ and $N(t)$ has the ...
0
votes
1answer
38 views

Intensity measure vs. Intensity function

Can someone please clarify the difference between an "intensity measure" and "intensity function" associated to point processes with an explanation including an example?
1
vote
0answers
32 views

Point process theory: Proof that strong mixing implies mixing.

Here's the problem: I'm working on a paper that says that strong mixing condition for stationary point processes implies the process to be mixing, but it never proves it (the paper is Ivanoff, Central ...
0
votes
1answer
56 views

Too small number out of a Point point process

I am trying to obtain $\mathsf{P}_{n}$, the probability that $n$ nodes appear in the ``intersection'' of two circles (with the same radius $r$) formed by two separate nodes, whose area is denoted as $\...
1
vote
2answers
37 views

Integral of shifted measure of set

Consider a measure $\mu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ such that $\mu(\mathbb{R}) < \infty$. In relation to showing stability of a point process, I need to show that for any bounded ...
1
vote
1answer
30 views

Especifying a measurable space for a homogeneos Poisson process.

I'm studying about the Poisson process (PP), and so far I can not find anything about the measurable space that the PP is defined. Then, I would like to know what is the measurable space for a ...
2
votes
0answers
91 views

Markovian Hawkes Process elementary proof

In the book An Introduction to the Theory of Point Processes I by Vere-Jones exercise 7.2.5 asks to show that the intensity of a Hawkes process with exponential intensity kernel is Markov. I found ...
2
votes
0answers
43 views

Stationary point process, bound $\mathbb{P}(N(0,\delta_0) \geq k)$

For a stationary point process, suppose we have the (terrible) bound $$\mathbb{P}\bigl( N(0,\delta) \geq 1 \bigr) \leq \sqrt \delta \quad \quad \text{ for each positive small $\delta$}.$$ Here $N(0,\...
1
vote
1answer
57 views

Poisson Process from independent non-identical exponential RVs

I know, I can define a Poisson Process using a sequence of i.i.d. exponential random variables, i.e. let $\tau_1, \tau_2, \tau_3, ... \sim \mathrm{Exp}(\lambda)$, then $T_i = \sum_{j=1}^I \tau_j$ are ...
1
vote
1answer
123 views

superposition of infinitely many poisson processes

I know that the superposition of two Poisson process with rates $\lambda_1$ and $\lambda_2$ is again a Poisson process with rate $\lambda_1+\lambda_2$. Thus this process has interarrival times ...
1
vote
0answers
11 views

Distribution of evaluated point process

Let $(X_n)$ a sequence of independent real random variables. Let $N_n = \sum_{k=1}^n \delta_{X_i}$ its point process and an interval $I\subset \mathbb{R}$. Then, $$N_n(I)= \sum_{k=1}^n \delta_{X_i}(I) ...
2
votes
0answers
59 views

Gibbs point process

I am reading in the book Spatial Point Patterns by Baddley et al. that "all finite point process models (under reasonable conditions) can be represented mathematically as Gibbs models". I couldn't ...
1
vote
1answer
53 views

Poisson process: finding probability of 1 count in an interval given that 0 counts happen in a subinterval

This was in my exam today and I'm not sure what's the correct answer. Let's say that the number of people that enter into a store in the interval $(0,t]$ (in hours) is a Poisson process where $30$ ...
1
vote
0answers
45 views

Doob-Meyer decomposition with respect to different filtrations

It is known that the Doob-Meyer theorem gives us a unique decomposition, $N(t)=A(t)+M(t)$ and the compensator part may conditional on a filtration $F_1$: $A(t|F_1)$. My question is: Does the Doob-...
0
votes
1answer
119 views

Homogenous Poisson Point Process to Binomial PPP

In my analysis, I am considering some nodes distributed as Homogenous Poisson Point Process (H-PPP) $\Phi$ with intensity $\lambda$. At a certain point during analysis, I need to focus on the ...
0
votes
1answer
129 views

How to test if an intensity function is a conditional intensity function?

I am trying to understand Hawkes process and I get that the conditional intensity is the expected number of events conditional on the past history. I have an intensity function that comes from my ...
1
vote
0answers
43 views

Random measures by random fields

Given a probability space $(\Omega,\mathcal{A},\mathbb{P})$, we have a random field $\{X_t\}_{t \in T}$, $T\subset S_1\times S_2$, for a measurable space $(S_1 \times S_2,\mathcal{A}_1\times\mathcal{A}...
0
votes
1answer
45 views

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 ...
0
votes
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 ...
0
votes
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: ...
5
votes
0answers
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 ...
3
votes
0answers
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 ...
1
vote
0answers
39 views

Poisson process uniquely identified proof: what is $\Gamma_r((\Theta ∟ A_i)^r)$?

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 $\...
1
vote
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 $\lambda_n(t,k)$. ...
0
votes
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 ...
1
vote
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?
1
vote
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 ...