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Questions tagged [random-measures]

For questions related to random measures (which can be defined as transition kernels or as random elements).

In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.

Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent.

To know more on this, check this link.

12 questions
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Probability that a jump belongs to a certain class

Let $N^a$, $N^b$ be two jump process with stochastic intensity process $(\lambda^a_t)_{t\in\mathbb{R}}$, $(\lambda^b_t)_{t\in\mathbb{R}}$ (the lambdas are $\mathcal{F}_t$ -adapted). Let $N$ defined by : $N_t := N^a_t + N^b_t$. Now define $\Delta N_t…
Sabrebar
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Concentration inequalities for random measures

For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality: $$\mathbb{P}\left(\left|\mu -\frac1n\sum_iX_i \right|\geq t\right)\leq…
Tyler6
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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
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Using Campbell's formula to find intensity measure $\Theta$

Problem Let $X_1, \ldots,X_n \sim K$ i.i.d random variables on $\mathbb{R}^d$, and let $$ \beta_{n,k} = \{ (x_1,\ldots,x_k) \in \{ X_1, \ldots,X_n\}^k \mid i \neq j, x_i \neq x_j \forall i,j \in \{1,\ldots,n\}\} $$ be the set of all $k$-tuples from…
Jacobiman
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How does Poisson thinning work?

I'm trying to understand how "Poisson thinning" can be used to construct a nonhomogeneous Poisson process. Here is what I got: Given a measure space $(E,\mathcal E,\alpha)$, a "Poisson random measure on $(E,\mathcal E)$ with intensity $\alpha$" is a…
0xbadf00d
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Spatial point process generation

Let $\varepsilon = \{\varepsilon_i\}_{i\in\mathbb{R}^d}$ be a collection of independent standard Poisson point processes on $\mathbb{R}^d$. That is, $\varepsilon_i$ are functions from a probability space to a space of simple counting measures on…
Albert Paradek
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Does jump measure of Brownian motion equals zero?

If $B=\{B_t\}_{t\ge{0}}$ is a Brownian motion adapted to $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t\ge{0}},P)$, let's define a Poisson random measure (or jump measure) as follows: Definition. Let $\textbf{B}_0$ be the family of Borel sets…
Roberto Palermo
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Campbell's formula for a random intensity

I try to obtain the cumulated intensity measure condition presented in the work from Schmidt, 2009, "Catastrophe Insurance Modeled by Shot-Noise processes" by applying Campbell's Formula. I have the following idea: we start with Campbell’s formula…
Anonymous
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Distribution of the "event locations" of a (nonhomogeneous) Poisson process

Let $(E,\mathcal E,\alpha)$ be a measure space. A Poisson random measure on $(E,\mathcal E)$ with intensity $\alpha$ is a transition kernel $\zeta$ from a probability space $(\Omega,\mathcal A,\operatorname P)$ to $(E,\mathcal E)$ such that if…
0xbadf00d
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Chebyshev's inequality for random measures

I'm currently reading "Iterating Brownian motions, ad libitum" (available here). Lemma 6 proves an equivalence criterion of vague convergence in the distribution of random measures. The statement and further ideas seem clear, but there is a…
user1047209
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Showing Integral wrt random possion measure has stationary and independent increments

Suppose $M$ is a Poisson random measure on $(E,\mathcal{E}) \equiv (\mathbb{R}_+\times\mathbb{R}^d, \mathcal{B}(\mathbb{R}_+\times\mathbb{R}^d))$ with mean measure $\nu\equiv Leb\times\lambda$. This means that: $\forall A \in \mathcal{E}$, the…
TomG
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Poisson Point Process as a random measure and finiteness of an integral

Suppose that $v$ is a Radon measure on $(0,\infty)$ and let $X$ be a Poisson Point Process on $(0,\infty)$ with intensity measure $v$. Let $Y:=\int xX(dx)$. In Theorem 24.17 of Probability Theory by A. Klenke (3rd version), the author, in showing…
Enrico
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