# Questions tagged [probability-measures]

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78 questions
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### How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?

Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...
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### Reference request: discretisation of probability measures on $\mathbb R^d$

Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e. $$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$ My concern is to approximate $\mu$ some $\mu_n$ that is ...
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### Conditional Expectation for $\sigma$-finite measures

Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure. I think it should be as follows: Let $(X,\mathcal{B},\nu)$ ...
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### Total variation and relative $\ell_\infty$ metric

Let $$D_{tv}(P,Q) = \frac{1}{2}\sum_{a \in A}|P(a)-Q(a)|$$ and $$D_{\infty}(P,Q) = \sup_{a \in A} \log \max\{\frac{P(a)}{Q(a)}, \frac{Q(a)}{P(a)}\},$$ where $P$ and $Q$ denote probability measures on ...
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### Weak-convergence of probability measures implies the convergence of the measure of a continuity set

Let $\Omega$ be a Polish space and $\mathcal{B}(\Omega)$ be its Borel $\sigma$-algebra. Let $\{\mu_n\}$ be a sequence of probability measures on $\mathcal{B}(\Omega)$ such that $\mu_n$ weak-converges ...
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### Upper bound for KL divergence on compact space

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability ...
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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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### Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups

Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2$ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (...
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### Time Interval of Existence of an SDE solution with Locally Lipschitz Drift

Consider the stochastic ODE $$dX = F(X)dt + dB$$ where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "almost ...
I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below. Suppose $(X_n,Y_n)$ are ...