This tag is for questions concerning point processes such as poisson point processes or any other point process.
Questions tagged [point-processes]
109 questions
11
votes
0 answers
Intuition behind the renewal equation
So. I've acquired the unenviable task of having to learn renewal theory on my own. I'm finding most of it to be pretty intuitive, except for one thing. The intuition behind the renewal equation has completely eluded me.
Every resource I've found so…
gogurt
- 2,274
7
votes
0 answers
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 for this requirement. That is, I would like to…
Henning
- 161
5
votes
1 answer
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 functional? I am a bit confused reading about it…
bilbo
- 329
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5
votes
0 answers
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 various authors stating this property but rarely…
Jfischer
- 1,073
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4
votes
2 answers
Convergence of subsequences of a stationary stochastic process
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 selection process, as $W_n = 1$ marks the occurrence…
nukelawe
- 85
4
votes
1 answer
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 severities are independent of times of occurrence…
Alexis Sandoval
- 376
4
votes
0 answers
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 recently published research articles here.
Are…
Abdulhameed
- 429
3
votes
1 answer
Existence of Malthusian parameter
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 $\mu(t)=\mathbb{E}[\eta(t)]$ and so…
elysian-peace
- 334
3
votes
1 answer
Concerns about the definition of Hawkes process
In the lecture notes I am reading about Point process, when we introduced the Hawkes process several expressions are given and I have some difficulty to understand properly what is the $Z_t$ (defined below). I tried to make my doubts clear by…
G2MWF
- 1,615
3
votes
1 answer
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…
user894121
3
votes
2 answers
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 distribution $F$. What is the long run percentage of…
TOMILO87
- 540
3
votes
1 answer
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 have some difficulties in understanding an elementary…
Gost91
- 479
3
votes
0 answers
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 find any reference for this statement. Any insight…
user2167741
- 535
3
votes
0 answers
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 with cells centered in the points generated by…
3
votes
1 answer
Minimum (Expected) distance between two points in a Poisson Point Process
If I have cellular base-stations distributed as a PPP $\Phi_C$ with $\lambda_c$ density. Then the pdf of distribution is well known i.e. $$P[N = n] = \frac{(\lambda_c\pi r^2)^n}{n!}e^{-\lambda_c\pi r^2}$$
My Take (I will be glad for any alternate…
SJa
- 849