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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1answer
53 views

Characteristic function say something about the expectation and variance [closed]

Show that if $\lim_{t \downarrow 0} (\varphi(t) -1) / t^2 = c > -\infty$ then $EX = 0$ and $E|X|^2 = -2c < \infty$. In particular, if $\varphi(t) = 1 + o(t^2)$, then $\varphi(t) \equiv 1$. Where ...
2
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1answer
48 views

Convergence in distribution of independent and uniform r.v.'s

This is the text of the problem: Let $\left(X_{j}\right)_{j\ \geq\ 1}$ be independent and let $X_{j}$ have the uniform distribution on $\left(-j,j\right)$. Show that $\lim_{n \to \infty}{S_{n} \over ...
3
votes
1answer
53 views

the density of the sum of $n$ random variables with uniform distribution on $(-1,1)$

Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density $$f(x)= \frac{1}{\pi} ...
0
votes
1answer
59 views

Linear Transformations $T$ and $S$ and their Characteristic Polynomials

My friends and I cannot figure out this proof. We have part (a) done, but weren't not quite getting part(b). We think we need a change-of-basis equation. Any advice?
1
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1answer
67 views

$\mathscr{B}$-matrix of T and Characteristic Polynomial

I'm having a difficult time trying to figure out this proof problem. Any advice on first steps? Let A be an $n\times n$ matrix satisfying the matrix equation $A^{n} + ...
1
vote
1answer
39 views

characteristic function characterize the distribution

Theorem: Let $\phi(t)=\int{e^{itX}dF_X}$ be a characteristic function of a random variable $X$. Then $\displaystyle \lim_{T \to \infty}\int_{-T}^{T}{{\frac{e^{-ita}-e^{-itb}}{it}}\phi(t)dt}=P(X\in ...
0
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1answer
43 views

What would be the simplified form of this expression?

I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows: $$ M_{X_n}(ju)= ...
1
vote
1answer
50 views

Show that $\frac{1}{n}\sum_{j=1}^{n}X_{j}$ is Cauchy distributed when the $X_{i}$ are all Cauchy

Let $X_{1}, \cdots, X_{n}$ be i.i.d. Cauchy random variables with parameters $\alpha=0$ and $\beta=1$. (That is, their density is $f(x)=\frac{1}{\pi\,(1+x^{2})}$, $-\infty < x < \infty$.) Show ...
0
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0answers
51 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
2
votes
1answer
56 views

How does what I did imply that $X$ is Normal $N(0,1)$?

Let $X$, $Y$ be i.i.d, that $X+Y$ and $X-Y$ are independent, and that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. Also, let $E\{X\}=0$ and $E\{X^{2}\}=1$. Show that $X$ is Normal $N(0,1)$. ...
2
votes
1answer
31 views

Show that $|1-\varphi_X (u)|\leq E\{ |uX| \}$

Show that $|1-\exp\{ix\}|^{2}=2(1-\cos x) \leq x^{2}$ for all $x \in \mathbb{R}$. Use this to show that $|1-\varphi_X(u)|\leq E\{|uX|\}$, where $\varphi_X(u) =E\{\exp(i\langle u,X\rangle)\}$ is the ...
0
votes
1answer
80 views

Let X, Y be i.i.d, X+Y and X-Y independent, show that the characteristic function E{exp(i<2u,x>)}= …

Let $X$ and $Y$ be i.i.d. Suppose further that $X+Y$ and $X-Y$ are independent. Show that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. What I tried to do was work backwards, starting with ...
0
votes
0answers
44 views

For X, Y real valued and independent, and X and X+Y having the same distribution, Y=0 a.s.

Let X, Y be real valued and independent. Suppose X and X+Y have the same distribution. Show that Y is a constant r.v. equal to 0 almost surely. Here's what I have so far: By the uniqueness of ...
2
votes
1answer
40 views

What is the meaning of $1_{a>b}$?

What would this mean: $1_{a>b}$ .. Based on the context, it could mean "$1$ if $a>b$ else $0$", but it's the first time I see it so help would be appreciated.
1
vote
1answer
198 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
0
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0answers
38 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
0
votes
1answer
29 views

Even numbered moments of N(0,1) using characteristic functions

Let $X$ be $N(0,1)$. Show that $E\{X^{2n+1}\}=0$ (Easy - calculate it directly using the definition of expectation, and you're taking the integral of an odd function over a symmetric interval, so =0), ...
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0answers
70 views

Characteristic function of an r.v. with finite variance and zero mean.

Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
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4answers
110 views

Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment

Let $X_{1},\cdots,X_{n}$ be independent, each with mean 0, and each with finite third moments. Show that $E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} ...
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0answers
216 views

Characteristic Function of a Double Exponential (Laplace) Distribution

Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$. Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$. ...
2
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0answers
88 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
1
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1answer
76 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
0
votes
2answers
45 views

Finding the characteristic ODE from a nonlinear PDE

I am studying for a PDE exam on Tuesday, and I am getting pretty confused about one specific type of problem and I am thinking that perhaps I am misinterpreting the correct procedure to follow. The ...
2
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1answer
92 views

Prove that $ \mathsf{E}[g(X)] = \int_{- \infty}^{\infty} G(t) \varphi(t) \, d{t} $.

Problem Let $ X $ be a real-valued random variable with characteristic function $ \varphi $. Suppose that $ g: \mathbb{R} \to \mathbb{R} $ satisfies $$ \forall x \in \mathbb{R}: \quad g(x) = ...
0
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1answer
144 views

Characteristic function

Question: Let $X_1$ and $X_2$ denote independent real-valued random variables with distribution functions $F_1$, $F_2$, and characteristic functions $\varphi_1$, $\varphi_2$, respectively. Let Y ...
2
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2answers
123 views

Characteristic function of $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$

Let X denote a real-valued random variable with an absolutely continuous distribution with density function $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$. Find the characteristic ...
2
votes
3answers
567 views

Solving a recurrence relation with the characteristic polynomial

Consider the sequence $\{a_n\}_{n=0}^\infty$ with $a_0 = 0, a_1 = 1, a_{n+2} = 6a_{n+1} - 9a_{n}$. Using the characteristic polynomial prove $a_{n} = n3^{n-1}$. So I really wasn't sure where to ...
1
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2answers
112 views

characteristic functions

I need to prove that if $\phi(t)$ if a characteristic function then so is $e^{\lambda(\phi(t) -1)}$ for $\lambda$ > 0 My problem is that I'm stuck at proving uniform continuity. Is it sufficient to ...
1
vote
1answer
52 views

Characteristic function problem

first time poster so be nice! Here's the problem: Let $\phi(t)$ be a characteristic function, then $e^{\lambda(\phi(t)-1)}$ is a characteristic function. Pretty stuck, any help appreciated!
0
votes
1answer
109 views

Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
0
votes
0answers
43 views

Find the characteristic equation of a recursive function

I want to determine whether the following recursive function is unstable; $$ x(t+1) = \left( wx+sx(t)^b \over w+x(t)^bs + (1-x(t))^b(d-s) \right) $$ Wikipedia is telling me that I want to have the ...
0
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0answers
13 views

multi-normal N(m,C): if C is not invertible, what's the impact?

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the definition of multi-normal distribution. In Appendix A, page 307 (sixth edition) it says: a random variable ...
0
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1answer
49 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
0
votes
1answer
284 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
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0answers
229 views

Characteristic functions in set theory

The book I am studying has the definition of a characteristic function as follows. Let $A\subseteq{X}$. Then $$\chi_A(x) = \begin{cases} 1, & \text{if $x\in{A}$} \\ 0, & \text{if ...
4
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1answer
445 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.
2
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0answers
212 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
1
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1answer
247 views

Conditional Characteristic Function

Given I know the joint characteristic function of the random variables $X,Y$ and the characteristic function of $Y$, is there a way to recover the characteristic function of $X|Y$ without inverting ...
2
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2answers
444 views

Step Function and Simple Functions

Definitions: Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here. It should follow that neither all simple functions are ...
8
votes
2answers
127 views

Does “independence” of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how ...
1
vote
1answer
34 views

$A_n \uparrow A \implies 1_{A_n} \uparrow 1_A$ pointwise?

Let $\Omega$ be the set. Let $1_A$ be the characteristic function on a set, i.e. it has value $1$ iff $x \in A$ and $0$ otherwise. Suppose There's a sequence of subsets $(A_n)_{n\geq 1}$ increasing ...
3
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0answers
78 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that ...
1
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1answer
68 views

Characteristic function of series

It is well known that if $X,Y$ are independent random variables on $(\Omega,\mathscr{F},P)$ with respective characteristic functions $\varphi_X,\varphi_Y$, then $\varphi_{X+Y}=\varphi_X\varphi_Y$. If ...
1
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1answer
130 views

Convergence in distribution of the log-Gamma distribution

Suppose $X$ has density $f(x)=\exp(kx-e^x)/\Gamma(k)$, $x>0$, for some parameter $k>0$. Then the moment-generating function of $X$ has the form $$ M_X(\theta)=\frac{\Gamma(\theta+k)}{\Gamma(k)}. ...
1
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1answer
184 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 ...
1
vote
1answer
144 views

Limit of characteristic functions

Let $\xi_1 ... , \xi_n$ be iid with $E \xi_i^2 < \infty $ what is $$ \lim _{n\rightarrow \infty} \varphi_{\bar{\xi}}$$ where $\varphi$ is the caracteristic function and $\bar{\xi}$ the mean of all ...
7
votes
1answer
230 views

Laplace transform of a random variable

My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
2
votes
1answer
60 views

Showing $\varphi(t)\neq 0$ when $\varphi$ is a characteristic function of an infinitely divisible distribution

Let $\varphi$ be a characteristic function of an infinitely divisible random variable. Show that $\varphi(t) \neq 0$ for all $t$. Sorry, I have no clue how to do it, because if the exponential is ...
1
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1answer
161 views

Inverse Fourier transform of characteristic function

Let $Q$ be a measurable set in $\mathbb{R}^2$ Let \begin{equation} 1_Q(\textbf{x}) = \left\{ \begin{array}{ll} 1 & \mbox{if $\textbf{x} \in Q$},\\ 0 & \mbox{otherwise},\end{array} \right. ...
2
votes
1answer
88 views

X,Y are independent RVs with known characteristic functions. Find P(X+Y=2).

X,Y are independent random variables with the following characteristic functions: $ \phi_X(\theta) = \frac{1}{4}e^{i\theta}+\frac{3}{4}e^{i2\theta} \\ \phi_Y(\theta) = ...