# Tagged Questions

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### “Direct” proof (without hypercontractivity) of equivalence of moments?

Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination ...
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### Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys $$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$ with additional condition that $\mathbb{E}X^k$ does not ...
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### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...
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### A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, ...
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### Inequality for the first Fourier level of a Boolean function

In the study of Boolean functions, the hypercontractive inequality enables one to bound from above the norm of $Tf$ by some norm of $f$, where $T$ is the noise operator depending on the noise ...
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### Number of lattice points in a random disk of radius r

Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
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### Fourier transform of distributions with non-standard test functions

This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation: Given a ...
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### Fourier Transform of measure on Banach Space (a question about Pontryagin Duality)

The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real): $\hat{\mu}: E^*\rightarrow \mathbb{C}$ defined by ...
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### Positive-Definite Functions and Fourier Transforms

Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite. ...
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### Good probability measues on $S^1$ reprented by a kernel

I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
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### Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
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### Can I relate the L1 norm of a function to its Fourier expansion?

I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
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### approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
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### Why do Littlewood-Paley projections behave like iid random variables

I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like ...
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