1
vote
0answers
38 views

I want to show $\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$

Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in ...
3
votes
1answer
69 views

Stable law and Levy distribution

A PDF (probability density function) f(x) is called a stable law if $f(y)=b\int_{-\infty}^{\infty}dx f(by-x)f(x)$ under appropriate values of b. Rewrite this equation in terms of characteristic ...
3
votes
0answers
75 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 ...
7
votes
1answer
226 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 ...
3
votes
2answers
322 views

Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
2
votes
1answer
92 views

Missing assumption? (Convergence of random variables and characteristic functions)

Here are two exercises from my Probability book ("Probabilidade: um curso em nível intermediário", by Barry James). (I have translated them from Portuguese to English and modified them a bit.) ...
4
votes
1answer
227 views

Combinations of i.i.d Inverse Chi-Square RVs and their characteristic functions

I am working on a few self-study problems in probability/measure theory and am stuck on characteristic functions. I have the following problem: Given: $X_1,\ldots,X_n$ are iid inverse chi-square(1) ...
4
votes
0answers
2k views

Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...