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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1answer
473 views

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 ...
3
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0answers
43 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 ...
1
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2answers
48 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] +...+\...
0
votes
1answer
25 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 ($z \in \mathbb{C}$) $$ ...
0
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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:...
2
votes
1answer
299 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 ...
3
votes
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
votes
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}^...
1
vote
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[ \...
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 $S_N$....
10
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1answer
2k views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
1
vote
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? ...
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 $v\neq 0$ ...
1
vote
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 $X_i$. ...
0
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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 ...
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 ...
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
vote
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 ...
2
votes
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 $\...
3
votes
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(\...
0
votes
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
vote
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 ...
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
votes
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
vote
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 ...
0
votes
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 = \...
0
votes
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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vote
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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vote
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
20 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
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 ...
2
votes
1answer
30 views

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. ...
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\...
5
votes
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.
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
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 ...