# Tagged Questions

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### Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
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### Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
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### DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
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### Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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### Convegence of regularized sequence in $L^2$

Let $(\rho_n)_{n \geq 0}$ be a standard regularizing sequence on $\mathbb R$. Let $P$ be a probability measure on $\mathbb R$ such that the sequence $(P*\rho_n)_{n \geq 0}$ is bounded in $L^2$. Then, ...
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### Pointwise Ergodic Theorem - one particular estimate

I am struggling with an estimate in a proof of the pointwise ergodic theorem discussed in T. Ward and M. Einsiedler: Ergodic Theory with a view towards Number Theory, which is left as an exercise. I ...
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### Maximum of measures over sets and functions

Let $(X,\mathcal A)$ be any measurable space and denote by $\mathrm b\mathcal A_1$ the set of all real-valued $\mathcal A$-measurable functions $f$ satisfying $\|f\|:=\sup_{x\in X}|f(x)|\leq 1$. Let ...
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### A concrete example of a non-pure-point measure

I am considering $P(\mathbb{R})$ the space of probability measures on the real line. We can regard this as within the space of continuous linear functionals on the space of continuous functions ...
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### Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
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### Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and suﬃciently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have;  E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
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### How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?

Let $X$ be a separable and locally compact metric space and let $P\colon X\times \mathcal{B}(X) \rightarrow \mathbb{R}$ be the transition probabability kernel of a homogeneus Markov chain on $X$. ...