Questions tagged [characteristic-functions]

Question 1

Consider the following family of normalized probability densities parametrized by the strictly positive integer $k$: $$ \begin{align} \begin{aligned} &f_k(x) = \frac{k}{\pi}\sin\left(\frac{\pi}{2k}...

Question 2

Let $A\subseteq \mathbb{R}$ be any set, and define its characteristic function as: $$\chi_A := \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{if } x \notin A \end{cases}$$ I need to prove ...

Question 3

Suppose that $X_1, X_2, \cdots ,X_n$ are i.i.d with $X_i \sim U([-\sqrt3,\sqrt3])$, $\Phi(t)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{t}e^{-\frac{x^2}{2}}dx$. Show that there exists $C>0$ such that $$\...

Question 4

Let $G$ be a finite group and $S\subseteq G$. Let $\rho$ be an irreducible representation of $G$ and $\chi$ be its (irreducible) character. Define $$\rho(S):=\sum_{s\in S}\rho(s),$$ $$\chi(S):=\sum_{s\...

Question 5

Let $\mu$ be a probability measure on $\mathbb{R}$ and let $\varphi$ be its characteristic function. Exercise 3.3.3 on Durrett's book about probability tells us that $$\lim _{T \rightarrow \infty} \...

Question 6

At the moment I am taking a measure-theory based probability course. In the previous homework assignment we were asked the following: Evaluate the limit $$\lim_{n\to\infty} \frac{1}{2^{n}} \int_{-1}^{...

Question 7

Consider the box function: $g_\epsilon=\frac{1}{2\epsilon}\chi_{[-\epsilon,\epsilon]}:\mathbb{R}\rightarrow \mathbb{R}$ for $\epsilon>0$. For a step function $t$, being a finite linear combination ...

Question 8

I want to evaluate an integral and to do that found a trick to integrate the primitive instead of the function (in my case, that really helps). But in the end, I always get 0, which should not happen. ...

Question 9

Background While studying the Poisson distribution, I came across the equation: $$ \mathbf{E}[\lambda\,g(X)] = \mathbf{E}[X\,g(X-1)], $$ which holds for a Poisson random variable $X \sim \text{Poisson}...

Question 10

Question: Let $X$ be a random variable taking values in $\mathbb{R}$ and let $F$ be its probability distribution function (that may not have a density). Denote the characteristic function as $\varphi(...

Question 11

This is a sequel of Ratio of cubic and quadratic form is approximately normal? Let be $x_{1},x_{2},..., x_{n}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. ...

Question 12

It is well known that the characteristic function $\varphi(\mathbf{t})$ of $\mathbf{x}\in\mathbb{R}^{n}$ following a spherical distribution is of form $$ \varphi(\mathbf{t})=\phi(\mathbf{t}^{\mathrm{...

Question 13

I am studying real analysis. I ended up with the following exercise: suppose that $f \in L^1(0, 1)$, $f \geq 0$ a. e. and suppose that there exists $c \geq 0$ such that $$\int_0^1 (f(x))^n dx = c, $$ ...

Question 14

I was taught that any orthogonal matrix must have eigenvalues of magnitude 1, i.e. $\lambda_i=\pm1,\forall i$. However, I am given the following matrix $$ M = \frac{1}{2}\begin{pmatrix} -1 & -1 &...

Question 15

I was attempting to prove the characteristic function formula $$Z_x(\lambda) = \sum_{n=0}^{\infty} \frac{i^n}{n!} \sum_{j_1, ..., j_n} \lambda_{j_1} ... \lambda_{j_n} \langle x(r_{j_1}) ... x(r_{j_n}) ...

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Questions tagged [characteristic-functions]

Known properties of these generalized Cauchy distributions

For a characteristic function, how to prove there is no subset A s.t limit of the function exists at only one point?

A Berry-Esseen-type inequality for uniform distribution

The sum of an irreducible representation over a subset of a finite group

Does the characteristic function of a measure tell us whether it contains an atom? [duplicate]

Evaluate a crazy limit (in the context of Probability)

convolution of step function with box function

Integration by parts and approximating a characteristic function seems to show that nearly every integral vanishes

Do the ODEs satisfied by characteristic functions have a probabilistic interpretation?

Sufficient Conditions for lower bound on absolute value of Characteristic Function

Ratio of cubic and quadratic form, as elementary symmetric polynomials, is normal?

Deriving the density of spherical distribution from its characteristic function

About characteristic function and their powers

Trace of an orthogonal matrix not equal to sum of eigenvalues

Proving a multivariate characteristic function formula

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