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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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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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30 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
24 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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30 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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29 views

Countable sum of point processes

Let $(\mathbb{X},\mathcal{X})$ be a measurable space. A point process is defined as a measurable mapping $$\eta : (\Omega, \mathcal{F}) \rightarrow (\textbf{N}, \mathcal{N}),$$ where $\textbf{N}$ ...
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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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22 views

Joint-intensity vs joint density of a point process

For a (simple) point process X, the joint intenisties are defined to be the functions (symmetric, locally integrable) $p_k: \Lambda^k \to [0, \infty)$ such that for any finte collection of disjoint ...
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15 views

Poisson point process representation

Let $\Pi: ( \Omega, \mathcal{F}, \mathbb{P} ) \rightarrow \mathbb{R}^d$ be a Poisson point process. We know that $\Pi_0=\{\left \| X \right \|, X\in \Pi\}$ is a Poisson point process on $\mathbb{R}_+$...
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22 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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13 views

Convergence of row superposition of null array to Poisson point process

Background: The limit of superposition of infinite number of independent point processes is a Poisson process under certain conditions. The conditions for the limit process to exist and be Poisson is ...
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Are all cluster point processes considered as inhomogeneous?

According to the definition in literature on spatial statistics (for example Spatial Point patterns: Methods and Applications with R, p. 137), point patterns for which the average density of points is ...
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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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23 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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21 views

optional integer-valued random measure v/s multivariate point process

I want to known what is the difference between an optional integer-valued random measure and a multivariate point process. By the formula $$ \mu(\omega;dt,dx)=\sum_{k\geq1}I(T_k<\infty)\epsilon_{(...
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1answer
42 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
48 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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30 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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24 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
126 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
26 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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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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Does it exist a known non-homogeneous point process with fixed number of points?

Having a grid $W=[0,1] \times [0,1]$ I'd like to generate a fixed number $k$ of points, distributed in $W$ according to the density function: $$ \rho(x,y) = \frac{e^{x+y}}{1 + e^2} \tag 1$$ I'm ...
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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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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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25 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
52 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
27 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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68 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
279 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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156 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
51 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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44 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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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
58 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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80 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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119 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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424 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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220 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 $\Lambda({\rm B})$ (...
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55 views

What is the space of all possible counting measures?

I have troubles understanding the definition of a random measure. Wikipedia says: If $f$ is some measurable function on $\mathbb{R}^d$, then the sum of $f(x)$ over all the points $x\in N$ can be ...
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1answer
234 views

Application of the Superposition Theorem for Poisson point processes

The problem below is Exercise 3.8 of Last and Penrose Lectures on Poisson Processes. I have been thinking for the better part of the day about it but couldn't write down even one insightful line. I am ...
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100 views

The (countable) sum of proper processes is a proper process

I am working with the script of Günter Last and Mathew Penrose Lectures on the Poisson Process (available online). Definition 2.4 We shall refer to a point process $\eta$ on $\mathbb{X}$ as a ...
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1answer
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How to compute the probability of $P(N_A = 1)$ considering an area $A$ in a Poisson point process?

Knowing that each point process (PP) is characterized by its void probabilities: $f(A)=P(N_A=0)$ as definition: the probability to have 0 points of the PP in the area $A$. And also, knowing that for a ...
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168 views

Related to the probability generating functional of Poisson point processes

A one dimensional poisson point process (PPP) $N$ has intensity measure $\Lambda(0,x)$ and we want to find the following $$E\left[\prod_{x\in N} e^{-\frac{s}{x}}\right]$$ using the definition of ...
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1answer
155 views

Integral with respect to an integral measure

This question originates from the definition of the Cox point process, but I suspect it might be a more general one. If we define $$Q(\cdot) = \int_{\mathcal M} P_{\Lambda}(\cdot)Q_{\Psi}(d\Lambda)$$ ...
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1answer
477 views

Integration with respect to a Poisson random measure

Let $N$ be a Poisson random measure (PRM) on a Polish space, $\left(X,\mathcal{B}(X)\right)$, and let $\tilde{\nu}$ be its mean measure. Then, let $f$ be any non negative and bounded function on $X$. ...
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1answer
75 views

Explain the orderliness of Poisson process

For an Orderly Poisson Process, events occur at distinct points and not simultaneously. However, the reverse is not necessarily true, i.e, even if the events occur at distinct points, the process may ...
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1answer
45 views

When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is ...
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Intuitive difference between Laplace functional of Poisson Point Process (PPP) and independently marked PPP

The Laplace functional of the Poisson Point Process (PPP) $\Phi$ with intensity measure $\Lambda$ on $\mathbb{R}^d$ for non-negative function $f(x)$ is: $$ \mathcal{L}_\Phi(f) = \exp\bigg\{-\int_{\...
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1answer
60 views

Why can we choose a sequence of points uniformly?

It recently came to my knowledge that, even though there is no way to uniformly choose a random real number (since the Lebesgue measure is not a probability measure), there is a canonical way to ...
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134 views

Simulating a homogeneous Poisson process with finite number of points on $\mathbb{R}^2$

I have to simulate a homogeneous Poisson Point Process on $\mathbb{R}^2$ with fixed number of points. Any hints as to how to do it would be helpful. I know that for a bounded region W we first ...