All Questions
Tagged with pr.probability measure-theory
219 questions with no upvoted or accepted answers
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
- 167
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Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
- 407
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Distance between two sample quantiles
Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
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Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user128095
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Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
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If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?
Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...
- 167
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216
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Hoeffding's lemma for unbounded r.v with bounded exponential map
Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: $$E[...
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Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
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145
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Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
- 215
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What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
- 19
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Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
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question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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Measure induced by function
It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-...
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Extension of probability measure from a finite algebra to sigma-algebra with countable many generators
I apologize for probably trivial question, I am far from this field.
If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\...
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98
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coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
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(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
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transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
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