# Tagged Questions

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### Ask for a good reference for the calculus involving singular continuous measure [migrated]

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
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### For what measures $\mu$ does $\mu*f\in L^{\infty}$ where $f$ is a given continuous integrable function?

I am trying to characterize all measures on $\mathbb{R}$ such that $$\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,$$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
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### Inequality for the tail of normal distribution function

Let $ะค(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt$ be the cumulative distribution function of the standard normal distribution. Numerical calculations suggest the following ...
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### Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
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### Looking for CDFs that I can integrate a particular transformation of

I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate $$\int_{-\infty}^t \lambda(a(x+b))dG(x),$$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
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### Has anyone seen this series?

I come across the following infinite series. $$\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for t>0 and a>0}.$$ In particular, I am interested in the case where $a=1/4$. ...
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### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set. \label{main12} C= \{x\in \mathbb{R}^d ~|~ ...
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### General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
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### Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
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### Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
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### Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by ...
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### Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
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### approximately linear functions

i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies $f(x-y)=f(x)-f(y)+const$ then it is necessarily linear. are there any general ...
consider the following mappings, G and T, $y(s) = $Gx$(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$ $z(s) = $Ty$(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$ where $0< x(s)\leq 1$ ,$r(s)<0$ , ...
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem. Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let \$\rho \in ...