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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36 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 ...
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1answer
22 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 ...
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5 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?
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6 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 ...
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22 views

Probability of condition being true over interval given probability of condition

I have a point process with time dependent rate $\lambda(t)$. Let me call intervent time $\tau$ the temporal distance between two consecutive events. I know the probability that no events occur in ...
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31 views

Prove $N$ is a nonhomogeneous Poisson process with local intensity $\lambda(t)$.

Assume $N$ is a point process on $[0,\infty)$, and let $N(t)=N\left( [0,t]\right).$ Suppose $\{N(t), t\geq 0\}$ satisfies $N(0)=0$, $N$ has independent increments and \begin{align*} P[N\left((t,...
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8 views

What is the relation of this result to thinning? Construct a second probabilistic proof using thinning of a Poisson process.

If $\{E_{n}\}$ are idd exponentially distributed with parameter $\alpha$, $N$ is geometric with $$P[N=n]=pq^{n-1}, \quad n \geq 1,$$ and $N$ is independent of $\{E_{n}\}$. Here I show that $$\sum_{n=1}...
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1answer
69 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 ...
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30 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 ...
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26 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 ...
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36 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 ...
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46 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 ...
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1answer
25 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?
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27 views

Concentration Inequalities for point processes

I'm looking for some references in Concentration Inequalities on the counting random variable $ N(t) $ for Hawkes and Poisson (temporal) point processes. Could you direct me to some? I haven't ...
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19 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 ...
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19 views

Questions about Cox Process and spatial patterns

I am just learning about the Cox Process and have a few questions that I'm confused about. I have been reading that a Cox process is a Poisson process where the intensity parameter is random. So my ...
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1answer
48 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 $\...
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2answers
22 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 ...
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1answer
25 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 ...
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53 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 ...
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40 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,\...
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1answer
44 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 ...
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1answer
80 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 ...
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8 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) ...
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51 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 ...
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1answer
29 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 $30$ ...
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39 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-...
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1answer
94 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 ...
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1answer
111 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 ...
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0answers
39 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}...
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1answer
34 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 ...
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1answer
196 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 ...
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1answer
76 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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171 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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53 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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37 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 $\...
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57 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 $\lambda_n(t,k)$. ...
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1answer
65 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 ...
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1answer
38 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?
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70 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 ...
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1answer
554 views

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^...
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2answers
181 views

Integral of a sum dependent on the variable of integration

Imagine I have a process given by SDE $$ d\lambda_t = \kappa (\lambda_\infty - \lambda_t)dt + \delta_{1} dN_t $$ where $\lambda_\infty$ is a constant and $N_t$ is a poisson counting process. Solving ...
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1answer
58 views

Wondering how to get this analytical solution of $\text{E}\big(\log(f)\big)$, $f\sim$ Normal Distribution

I am reading variational inference for gaussian process modulated poisson processes and find the result (19) is unclear about its source. I am wondering how they get that. The equation is shown here \...
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51 views

From a TSP to a Minimal Euclidean Matching by Removing Edges

Both the optimal tour though 30 Euclidean points and a perfect matching constructed by removing every other edge, are displayed below: Is the matching minimal? If not, why not? What operation is ...
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79 views

Martingale estimating, Confidence interval

Halloo People, i must create confidence intervals for Martingale estimators. For Processes $B_{t}(a,k)=\int_{0}^{t}a_{s}(k)1_{(E[\lambda_{s}(k)]>0)}ds$ we have estimators $$\hat{B}^{n}_{t}(a,k)=\...
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1answer
71 views

Questions about a solution to a point process exercise

I have questions regarding the solution to this exercise: Exercise: Let $\eta$ be a stationary simple point process with intensity measure $\gamma \,\mathrm{d}x$ on $\mathbb{R}, \gamma >0$ such ...
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91 views

Computation of intensity measure for proper point processes

I am stomped by the following exam preparation question Problem: Let $\eta = \sum_{i=1}^\kappa \delta_{X_i}$ be a proper point process on some measurable space $( \mathbb{X}, \mathcal{X})$ with ...
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162 views

Proving that a process is a Poisson Point Process

I'm stuck proving the following proposition: Let $\{E_i\}_{i\geq 1}$ be i.i.d. exponential random variables on $[0,\infty)$ with parameter $1$: $P(E_i > x)= e^{-x}, x>0.$ Let $\Gamma_{n} =\sum_{...
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550 views

Conditional inter-arrival times for Poisson and Renewal Process

Assume $X_1,X_2,\ldots$ are positive inter-arrival times of a renewal process with pdf $p(x)$, i.e., the $k^\text{th}$ arrivals occurs at $\sum_{i=1}^k X_k$. What is the pdf of inter arrival times ...
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253 views

Intensity of Poisson point processes and its relation with probability density function (PDF) of nodes locations

Suppose $\Phi$ is a Poisson point process with intensity $\lambda(x)$. Then, for a given compact set B we have $\Lambda({\rm B})=\int_{\rm B} \lambda(x) \rm{d} x$. I know that $\Lambda({\rm B})$ (...