# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...