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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Why is the Sine Kernel admissible as the kernel of a DPP? [closed]

The Sine DPP is given by the kernel $$K(x,y)=\frac{\sin(\pi(x-y))}{\pi(x-y)}$$ and is a well-known example of a Kernel for a determinantal point process. However, this kernel is not always positive. ...
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Counterexample for Mecke equation in higher dimensions

I am currently reading the book Lectures on the Poisson Process by Gunter Last and Mathew Penrose. (The book can be found here.) I have a question about an exercise in the book's 4th chapter (Exercise ...
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is every counting Process a non homogeneous Poisson process?

I'm working on a counting process and trying to prove that this counting process is a Poisson point process (non homogeneous). I have the 3 conditions : $N(t)>0$ $N(t)\in \mathbb N,\forall t \in \...
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Sum of Point Processes is again a Point Process

I currently work on "lectures on the poisson process" (written by Last and Penrose). I struggle with an exercise in the book, more explicit exercise 2.3 It States that for a sequence of ...
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Proving Hawkes intensity has all the moments.

Let $N_t$ be a point process whose intensity follows Hawkes: \begin{align*} \lambda_t = \mu_t + \phi * dN_t, \end{align*} where $*$ is a convolution opeartor, $\phi * dN_t = \int_0^t \phi(t-s)dN_s$, ...
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What does this mean heuristically speaking $\lambda_t(i) = P(dN_t(i) = 1 | G_t)$

The question above relates to point processes and their internal history which is $G_t$. For me it is important to understand what the term on the right side means. I took this equation from a book ...
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Palm distribution of purely atomic random measure

I am struggling on Example 13.1(b) in the second volume of "An Introduction to the Theory of Point Processes" by Daley and Vere-Jones. Let $\xi = \sum_{j=1}^n \kappa_j \delta_{x_j}$ a random ...
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What is a Mark Variogram in point process?

I would like explanations about what is a mark variogram. I saw that Besag's L and Ripley's K functions where great agregation analysis tools, but i still wasn't able to wrap my mind around what is a ...
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Supremum of a random walk with random number of children in each generation. Recommend references

I have quite an interesting question, that I am pretty sure there is a lot of research already done about this, but I don't know in which field. Here are three different interpretations of the problem,...
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How to compare Poisson Point Process, ARIMA and LSTM?

I'm trying to compare three forecasting techniques: A stationary stochastic Poisson-GEV : where the rate of occurrence of the events is given by a Poisson process and, it's intensity is given by a ...
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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....
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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 ...
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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 ...
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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-...
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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\...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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1 vote
1 answer
60 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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1 vote
2 answers
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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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1 vote
1 answer
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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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131 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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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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1 vote
1 answer
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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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1 answer
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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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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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2 votes
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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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1 answer
72 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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1 vote
0 answers
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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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1 answer
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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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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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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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1 answer
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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 ...
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1 answer
289 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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1 answer
151 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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187 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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3 votes
0 answers
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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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