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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44 views

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 ...
0
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1answer
26 views

Khinchin's Law of Large numbers proof unclarity.

This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\...
2
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1answer
42 views

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 : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
0
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0answers
20 views

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 ...
1
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0answers
13 views

How to compute the $PDF_{X}(x)$ of $X$ if it cannot be Fourier inverted from the characteristic function $CF_{X}(z)$?

I have a positive random variable $X>0$. I have to compute the probability density function $$PDF_{X}(x)$$ I can compute in closed-form the extended characteristic function ($z \in \mathbb{C}$) $$ ...
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0answers
17 views

An identity relating the unknown $CDF_{X}(x)$ of $X>0$ and the known characteristic function $CF_{X^2}$ of $X^2$

I have a positive random variable $X>0$. I don't know that much about its distribution and I have to compute the cumulative distribution function $$ CDF_{X}(x) = Prob(X\leq x) $$ Other definitions:...
3
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0answers
29 views

Can inversion integral of characteristic functions on a finte interval be bounded?

For a real-valued uni-variate r.v. $X$, with pdf $f(x)$ and absolute integrable cf $\varphi(t)$, we have the following transform:$$2\pi f(x)=\int_{-\infty}^{\infty}e^{-itx}\varphi(t)\,dt.$$ However, I ...
0
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1answer
26 views

Counting solutions by estimating Fourier coefficients

In W. T. Gower's essay The Two Cultures of Mathematics, he mentions the following as an example of a 'general principle' in combinatorics: "If one is counting solutions, inside a given set, to a ...
2
votes
1answer
25 views

Continuity of a characteristic function of a translated set

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, ...
2
votes
1answer
23 views

Are derivatives of a characteristic function bounded?

Let $X$ be a real valued random variable with cdf $F(x)$ and characteristic function $\varphi(t)$, and suppose that $E[|X|^n]<\infty$ for some $n$. Then we know $$\varphi^{(k)}(t)=i^k\int_{-\infty}^...
2
votes
1answer
301 views

Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices ...
2
votes
3answers
27 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
40 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 $v\neq 0$ ...
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0answers
14 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
74 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[ \...
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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
votes
0answers
26 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
30 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
64 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
57 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
48 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
33 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
28 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
41 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 $\...
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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. $0$...
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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
21 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}...
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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
30 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
votes
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)|...
0
votes
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
82 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
59 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
36 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
vote
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
votes
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 ...