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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2
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3answers
26 views

Characteristic Function of a Conditioned Random Varaible

Let $(X_j)$ be iid random variables and $N \sim \mathrm{Poisson}(\lambda)$, independent of $X_j\, \forall j$. Define $S_n:=\sum_j^n\,X_j$ and consider $S_N$. Find the characteristic function of $S_N$....
1
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0answers
36 views

Characteristic function of a lattice distributed random variable

Let $X$ be a random variable. $X$ is called lattice distributed if there exist real numbers $a, b$ such that $P(X \in a +b\mathbb{Z})=1$. Show that $X$ is lattice distributed if there exists $v\neq 0$ ...
1
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0answers
12 views

Characteristic function of triangular distribution over $[0,2]$.

Here it is shown that if $X_1,\dots ,X_n$ are iid random variables, then the characteristic function of $S=\sum_{i=1}^nX_i$ is the product of the respective characteristic functions of the $X_i$. ...
1
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1answer
72 views

Kolmogorov's Truncation Lemma (ii)

Probability with Martingales: How exactly do we have the part in the $\color{red}{\text{red}}$ box? What I tried: $$E\left[ \sum_{n=1}^{\infty} 1_{|X| > n} \right]$$ $$ = E\left[ \...
0
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0answers
41 views

How can this be a characteristic function if it's not continuous

Consider the following probability density function $f(x)$ \begin{cases} 0 & x<-1 \\ 1+x & z\in[-1,0] \\ 1-x & z\in[0,1] \\ 0 & x>1 \end{cases} Then the ...
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0answers
16 views

Do characteristic functions characterize the independence of random variables? [Solved] [duplicate]

It is well known that the probability density function characterizes the independence of random variables in the following sense. $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$ where $f$ is ...
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0answers
19 views

Random variable with characteristic function cosine

So I am searching for a Random Variable $X$, such that $\varphi_X(t)=\cos(t)$. I know how to choose $X$ such that $\varphi_X(t)=e^{it}$ and $\varphi_X(t)=e^{-it}$. Does this help me? How can I put ...
2
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0answers
24 views

$e^{\varphi -1}$ characteristic function

So I am trying to figure out whether $e^{\varphi-1}$ is a characteristic function given that $\varphi$ is. I know that linear combinations of characteristic functions and the real part of a ...
1
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1answer
28 views

Complex Solids of Revolution

I know that to compute a solid of revolution of a function $f(x)$ rotated around the $y$-axis, one method we can use is the "shell" method. For example, $f(x)=1/4x^2\in [2,4]$, rotated around the $y$-...
1
vote
2answers
56 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
votes
1answer
47 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}] =\int_{\Omega}e^{itX(\...
1
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1answer
45 views

Characteristic functions and tightness of Uniform and Geometric distribution.

If $X_n$ has a $\mathrm{Uniform}(0,n)$, $Y_n$ has $\mathrm{Geometric}(1/n)$ 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? ...
0
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3answers
45 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 {d^2(y?z)}...
3
votes
1answer
47 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 $\...
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0answers
31 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
27 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
25 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
29 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 $\...
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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
28 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
votes
1answer
28 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
36 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 = \...
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0answers
40 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: $$y_t-5y_{t-1}+6y_{t-2}...
2
votes
1answer
23 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. $0$...
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
votes
1answer
17 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 $$a_{11}(x,y)u_{xx}+2a_{12}(x,y)u_{xy}+a_{22}...
0
votes
1answer
29 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}] = \frac{1}{\tau}\int_{...
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0answers
28 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
vote
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
votes
0answers
68 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)|...
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
79 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
votes
3answers
58 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}\\ &=\lim_{n\to\...
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
33 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
69 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 the ...
2
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0answers
41 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
votes
1answer
37 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 $\{X_n:n\ge1\...
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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
votes
0answers
42 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
vote
1answer
38 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
votes
0answers
73 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
votes
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 $\frac00$ can ...
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
votes
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
vote
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
votes
1answer
38 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 the ...
1
vote
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} &...