4
votes
2answers
98 views

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 ...
4
votes
0answers
76 views

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 $f(x)$, ...
2
votes
2answers
270 views

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, ...
8
votes
0answers
226 views

Uncertainty principle in Entropy terms

Math Questions: Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm $ ||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2}, $ and Fourier transform $ (F\psi)(\xi) = ...
1
vote
2answers
194 views

Berry Esseen inequality for multidimensional distributions

The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary $$ \sup_{t \in ...
3
votes
1answer
588 views

karhunen-Loeve expansion of Poisson process

Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda ...
3
votes
1answer
232 views

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 ...
10
votes
3answers
1k views

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 ...
1
vote
1answer
612 views

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 ...
0
votes
1answer
573 views

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 ...
7
votes
2answers
2k views

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. ...
0
votes
2answers
171 views

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 ...
11
votes
1answer
1k views

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 ...
5
votes
2answers
2k views

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 ...
7
votes
2answers
2k views

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 $X$ ...
12
votes
5answers
1k views

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 ...
2
votes
2answers
566 views

Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform. More concrete case, let $f:\mathbb{R} ...
1
vote
1answer
547 views

Decoupling lemma for the Lambda(p) problem

I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...