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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2answers
41 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
3
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1answer
36 views

Characteristic functions of random variables are non-negative definite

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X:\Omega\to\mathbb{R}$ a random variable. How to prove that the characteristic function $\varphi_X(t) = E[e^{itX}] ...
1
vote
1answer
39 views

Characteristic functions and tightness of Uniform and Geometric distribution.

If $X_n$ has a $Uniform(0,n)$, $Y_n$ has $Geometric(\frac{1}{n})$ and $Z_n$ has $\frac{1}{n}Y_n$ distribution how would you show whether or not each one is a tight sequence or not? Additionally how ...
0
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3answers
42 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
3
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1answer
46 views

Help with understanding the proof for: $AB$ and $BA$ have the same characteristic polynomial (for square complex matrices)

I saw many proofs but they all use advanced techniques and are impossible to understand. I'm looking for a proof that $AB$ and $BA$ have the same characteristic polynomial for any square matrix over ...
1
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0answers
29 views

Find the characteristic function of Y where Y|X=x $\in N(0, x)$ with X $\in Po(\lambda)$ [closed]

In particular $Var(Y|X=x)=x$. I think the solution should be the characteristic function for the $N(0,\lambda)$ distribution i.e. $\varphi_y(t)=e^{-\frac{1}{2}t^2\lambda}$ but I can't figure out why ...
1
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1answer
19 views

Characteristic function of discret random variable

I try to show the following: Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random ...
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2answers
22 views

Finding eigenvalues of a $3\times3$ matrix with Laplace expansion

Currently working on problem for a linear algebra class, but having a difficult time grasping eigenvalues. Here are the steps I'm doing: $$A=\begin{bmatrix}-5 & 1 & 0 \\ 0 & -4 & 3 \\ ...
1
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1answer
26 views

Strong equivalence between Lévy’s metric and a topologically equivalent metric

Let $\mathscr B$ be the Borel $\sigma$-algebra on $\mathbb R$ and let $\mathscr P$ denote the set of all probability measures on the measurable space $(\mathbb R,\mathscr B)$. Lévy’s metric on ...
0
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2answers
35 views

Find the sequence defined by the recurrence equation $x_{n+1} = 4x_n − x_{n−1}, (n ≥ 1)$

Find the sequence defined by the recurrence equation $x_{n+1} = 4x_n − x_{n−1}, (n ≥ 1)$ with $x_0 = 1$ and $x_1=2$. Find an odd prime factor of $x_{2015}$. I've found the characteristic equation to ...
2
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1answer
25 views

Characteristic functions proof… $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$

I am trying to prove Lemma 4.1 (1) from Olav Kallenberg's Foundations of Modern Probability: $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$ From context, I believe that ...
2
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1answer
27 views

How to compute the characteristic function

Let $X_n$ be a sequence of identically distributed, independent random variables and let the distribution of $X_n$ be $\mu_{X_n}(\{1\}) = a$ and $\mu_{X_n}(\{-1\}) = 1-a$ for 0 < a < 1. ...
0
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1answer
33 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = ...
0
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0answers
37 views

Stability and characteristic roots of difference equations

I often hear "For a process to be stable its characteristic roots or poles must be outside the unit circle (for casual process)". All right, consider the recurrence relation: ...
2
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1answer
22 views

What is the eigenvalue for characteristic polynomial $(1+\lambda^2)\lambda$?

What is the eigenvalue for characteristic polynomial $(1+\lambda^2)\lambda$? Is it $0,i,-i$ or $0,i,i$?
2
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0answers
33 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then ...
1
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2answers
30 views

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. ...
1
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1answer
24 views

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 \\ ...
0
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1answer
16 views

How to reduce into canonical form

Determine the type of the following equation and reduce the PDE to its canonical form $u_{xx} + 4u_{xy} + 4u_{yy} + u = 0$. We consider pdes in the form ...
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1answer
27 views

Finding a characteristic function of an exponential pdf

My pdf is defined as follows: $$f_X(x) = \frac{1}{\tau} e^{-x/\tau}$$ At first I started finding the characteristic function like so: $$\hat{f}_X(\xi) = \mathbb{E}[e^{i\xi X}] = ...
1
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0answers
27 views

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 ...
1
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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}} \ ...
5
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0answers
66 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} ...
0
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1answer
32 views

Given the characteristic equation of A.Find equation for B.

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 ...
5
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1answer
63 views

A sum of a random number of Poisson random variables

in my probability class I was given this question on which I am stuck concerning a sum of random number of Poisson random variables: Let us define the countable set of independent random variables ...
0
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3answers
57 views

What is the integral of the function $f = \chi_{[0,\infty)}e^{-x}$

I have $$\int_{\mathbb R} f\,d\mu = \lim_{n\to\infty}\int_{[0,n]} f\,d\mu$$ So $$\begin{align}\int_{\mathbb R}f\,d\mu &= \lim_{n\to\infty}\int_{[0,n]}\chi_{[0,\infty)}e^{-x}\\ ...
2
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3answers
57 views

$\exp(-t^4)$ is not a characteristic function

I'm looking for the answer to the problem in the title. I know it comes from the Marcinkiewicz theorem, but I need formal proof of it.
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0answers
32 views

Example for sum of dependent random variables of which the characteristic function can be factorized.

The characteristic function of two independent random variable can always be factorized, but the opposite is not true: I would like to construct an example, where the characteristic function can be ...
1
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1answer
64 views

Characteristic function

Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e, $$P\{X_1=1\}=1-P\{X_1=0\}=p.$$ The random variable Y is independent of ...
2
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0answers
37 views

Are the following functions characteristic functions of a random variable?

Let be $X$ an arbitrary random variable with characteristic function $\varphi(t)$. Prove that $$\psi(t)=\frac{1}{2-\varphi(t)},$$ and $$\chi(t)=\int_{0}^{\infty}\varphi(st)e^{-s}ds$$ are also ...
2
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1answer
34 views

Characteristic functions of infinite dimensional random elements

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 ...
1
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0answers
23 views

characteristic method for wave equation in a non-uniform string

wave equation in a non-uniform is $u_{tt}=c(x)u_{xx}$, $c(x)=1$, when $0<x<1/2$, $c(x)=2$, when $1/2<x<1$, Does $u(x,t)$ has the form like $u(x,t)=f(x-t)+g(x+t)$ when $0<x<1/2$, ...
4
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0answers
41 views

Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on a square lattice, where the initial configuration is $n$ chips on a single ...
1
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1answer
37 views

Characteristic functions, show that $\mathrm{E}[X_1 + X_2 +…+X_n]^3 = \mathrm{E}[X_1^3] +…+\mathrm{E}[X_n^3]$ when $\mathrm{E}[X] = 0$

I want to show that for independent variables X_1, X_2,...,X_n, with expectation 0 and finite third moments, $$\mathrm{E}[X_1 + X_2 +...+X_n]^3 = \mathrm{E}[X_1^3] + \mathrm{E}[X_2^3] ...
2
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0answers
70 views

Convergent sequence of characteristic functions with continuous integrable limit.

Suppose $\varphi_{n}$ is a sequence of characteristic functions converging pointwise to a function $\varphi$ which is continuous and absolutely integrable. I want to show that $\varphi$ is the ...
0
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1answer
30 views

Prove that $u^{-1}\int_{-u}^u(1-\varphi(t))dt\to 0$ as $u\to 0$

How to prove that $u^{-1}\int_{-u}^u(1-\varphi(t))dt\to 0$ as $u\to 0$ $\varphi(t)$ is the characteristic function of a random variable, if $u\to 0$ then $\varphi(t)\to 1$, so one gets $\frac00$ ...
0
votes
1answer
25 views

Calculate characteristic function

$p(n)=(1-r)^2nr^{n-1},n=1,2,...$ $f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$ the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not ...
0
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1answer
25 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
1
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1answer
27 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
0
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1answer
36 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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0answers
51 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} ...
0
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1answer
23 views

Probability with Indicators Textbooks

I am new to using indicator functions (although I am quite familiar with undergrad-level probability and what an indicator function is). I am trying to relearn probability using indicator functions ...
3
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1answer
60 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
7
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0answers
65 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega ...
0
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0answers
23 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
1
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1answer
92 views

Characteristic function and moment generating function: differentiating under the integral

In order to justify the interchange of the derivative and integral when differentiating a characteristic function, one can use the dominated convergence theorem: $$\frac{d}{dt} \int e^{itx} P(dx) ...
2
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0answers
54 views

Proofs of some characteristic function properties

I saw these properties in the wikipedia page but I was unable to prove them. I had an idea to construct a random variable with the desired characteristic function, yet I haven't managed to that so ...
2
votes
1answer
81 views

Does $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ imply independence of $X$ and $Y$? [duplicate]

It shouldn't, but I am blanking on a counterexample. ETA: Note that the $t$ is shared on both sides - which differentiates this from this question. Similarly $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ implies ...
2
votes
0answers
34 views

Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n), $ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
0
votes
1answer
31 views

Calculating characteristic function of random variable

I would like to calculate the characteristic function of $Z_{\beta, n}=(1-\beta^2)^{1/2}\sum_{k=0}^n\beta^kX_k$, where $X_i$ are independent random variables with $P(X_i = 1)=P(X_i=-1) = 1/2$ and ...