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Mathematics

Questions tagged [characteristic-functions]

Questions about characteristic functions, of a set (which gives 1ドル$ if the element is on the set and 0ドル$ otherwise) or of a random variable (its Fourier transform). Do not use this tag if you are asking about the method of characteristics in PDE or the characteristic polynomial in linear algebra.

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1 answer
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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 ...
1 vote
0 answers
83 views

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

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\...
0 votes
0 answers
35 views

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} \...
9 votes
1 answer
778 views

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}^{...
0 votes
1 answer
47 views

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,ドル being a finite linear combination ...
1 vote
1 answer
121 views

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. ...
3 votes
0 answers
84 views

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

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(...
2 votes
0 answers
75 views

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 $\sigma=1$. ...
3 votes
2 answers
167 views

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{...
1 vote
1 answer
69 views

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

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 &...
1 vote
1 answer
59 views

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}) ...
0 votes
0 answers
64 views

I have a problem where I want to integrate over is \begin{equation} \int{}I_{\{f(t)\}}(s)n(s,t)ds. \end{equation} $f(t)$ is a continuous function and at any $t$ the set $\{f(t)\}$ is a single point. I ...

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