# Tagged Questions

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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### Infinitesimal generator and stationarity

The following question is bothering me. I think it is probably known but I cannot find any reference... Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal ...
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### Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$. ...
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### Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions

Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
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### Measurability of integrals with respect to different measures

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
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### Fell topology versus vague topology for representing random sets

I'm trying to better understand the consequences of representing a random set as a Random element in the space of locally finite closed sets under the Borel sigma algebra generated by the Fell ...
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### Total variation, Wasserstein, and Prokhorov metrics on countably infinite discrete spaces

Total variation, Wasserstein, and Prokhorov generate the same topology on the space of probability measures on a finite and discrete space. I'm curious about a countably infinite space. When do ...
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### Relative compactness and convergence in probability

Convergence in probability of a sequence of random variables $X_1,X_2,\dotsc$ implies that every subsequence has a further subsequence that converges almost surely. Superficially, it seems that this ...
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### Influence of independent variables on boolean functions?

Suppose a simple connected graph $G$ where its vertices are assumed to be independent. An event with uncertainty corresponds to each vertex. My instructor guides me that even though the vertices (...
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### Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal? (functional integrals in probability theory) Clearly my question looks at the same time fairly ...
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### Modify Process to a Semimartingale

The original post is from mathstackexchange According to some difficulties, i decided to ask here again. Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a ...
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### How to generalize normal number theorem

The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
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Let $X_t$ be a random process such that \begin{eqnarray} X_1 &=& 0\\ X_t &=& X_{t-1} + \left\{\begin{array}{ll} A_t, & X_{t-1} \geq 0\\ B_t, & X_{t-1} < 0 \end{array}\...
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### Limit theorem : reproduce a proof with an adaption from discrete to continuous time

Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes". The setup is as follows: We have a Levy-Process $X_t$ fullfilling the CLT \begin{align} \sqrt{t}(X_t/t-E(...
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### Generalizing the law of large numbers to multiple sets of samples

The law of large numbers says that if I sample $n$ points independently from a probability density function $f$, then the number of points lying in a neighborhood of a point $x$ with area $\epsilon$ ...
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### Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1).$$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
### An inequality on the simplex involving $x^x$
Is anything known about the behavior of the function $$f(x)=\prod_{i=1}^n x_i^{x_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum_{i=1}^n x_i=1, x_i\geq0\}$? I ask because I have ...