# Tagged Questions

Questions about characteristic functions, of a set (which gives $1$ if the element is on the set and $0$ 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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### Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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### Why is $|\varphi(t)|$ not necessarily a characteristic function?

I came across the following statement in a book: If $\varphi(t)$ is a characteristic function, then $|\varphi(t)|$ is not necessarily a characteristic function. Here's my argument: By Bochner’s ...
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### Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
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### Characteristic function of standard normal distribution using this method.

Lets have $f(t)$ be this characteristic function. I am told that $f'(t)=-t \cdot f(t)$ and that this can be proven, I found using partial integration and the dominated convergence theorem. I am aware ...
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### How would I find the characteristic equation of this Recurrence Relation?

Find and solve a recurrence relation for the number of $n$-digit ternary sequences with no consecutive digits being equal. Since for ternary, meaning only $3$ possible entries for each space, e.g. $0$...
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### How can I make sure that the classical way of calculating the characteristic function of an exponential holds?

Given $f(x)=\lambda e^{-\lambda x}$, I want to find $\phi(t) = E(e^{itx})$ (characteristic function). Classical way: \begin{align} \phi(t) &= \int_0^{\infty} e^{itx}\lambda e^{-\lambda x} dx \\ ...
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### Difference between the characteristic function of a sum of independent vectors and a Gaussian

In the book "Sums of Independent Random Variables" by Petrov the following lemma appears in page 109, in preparation to prove the Berry-Esseen inequality Let $X_1,...,X_n$ be independent random ...
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### Showing the sum of random variables are Linnik-Distributed

Suppose $X_{1},X_{2}$... are independent, identically Linnik(${\alpha}$)-distributed random variables and that N$\epsilon Fs(p)$ (First Success) and that N and $X_{1},X_{2}$...are independent. ...
I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that $\{X_n:n\ge1\... 1answer 2k views ### How to get PDF from characteristic function I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you. 1answer 49 views ### Analogous result for basic Central Limit Theorem Let$X_1, X_2, ..., X_k$be an independent and identically distributed random variables. Assume$\mathbb{E}(X_i^2) < \infty$for$1 \leq i \leq k$and$$\frac{X_1 + X_2 + ... + X_k}{\sqrt{k}} \ \... 0answers 69 views ### Properties of characteristic functions under statistical dependence Given random variables$X,Y,Z$,and$\phi(.)$denoting the characteristic function, I can see that the following is true when$Z$is independent of$X,Y$:$|\phi_{X+Z,Y} (t, s) − \phi_{X+Z}(t)f_{Y} (s)|...
Given that $\chi_A(x)=x^3-ax^2+bx-c$ Find $\chi_B(x)$ For: a)B=A-2I b)$B=A^2$ For a) would you put x+2 in for x in the $\chi_A(x)$. As Det(XI-B)=Det(XI+2I-A)= det((X+2)I-A) And b: Im not sure ...