Tagged Questions

2
votes
2answers
45 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
2
votes
1answer
74 views

Characteristic function converges pointwise

Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S
0
votes
1answer
52 views

Integral of characteristic and density function

Let X and Y be random variables (real valued) with density functions $f_X, f_Y$ and characteristic functions $I_X, I_Y$. How can we show that: $ \int_{-\infty}^{\infty}{I_X(y) f_Y(y) e^{-ity}dy} = ...
3
votes
2answers
50 views

A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
2
votes
1answer
52 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.) ...
1
vote
1answer
65 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
vote
3answers
111 views

Is this function - characteristic function of a random variable?

$\phi(t)= \begin{cases} 1,&\text{if $|x|< 1$;}\\ e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.} \end{cases} $ Can anyone help?
3
votes
1answer
128 views

Random variable with characteristic function $\large\frac{\phi(t)+\phi(-t)}{2}$ [duplicate]

Possible Duplicate: Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$ If $\phi(t)$ is the characteristic function of a random variable $X$, then $\Re(\phi(t))$ is ...
5
votes
1answer
82 views

Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
4
votes
1answer
133 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) ...
2
votes
1answer
146 views

Applying an inversion technique to Characteristic Functions

I am struggling with this concept (self-study). Could someone show me how to explicitly apply the inversion formula for these examples? I am working through about 15 examples, but these 3 seemed ...
2
votes
1answer
76 views

Characteristic functions (Statistics)

I would greatly appreciate any help with this problem. If $f_1, f_2 , f_3$ are three characteristic functions (in Statistics, e.g $E(\exp(itX)))$ such that $f_1*f_3=f_2*f_3$ for all $t$ and we are ...
2
votes
1answer
498 views

Characteristic function of a sum of Uniform random variables

Suppose I have $S=U_1+U_2+\dots+U_n$ where $U_i$ are distributed Uniform$[-1,1]$. I am trying to show a couple of things. First, what is the characteristic function. I can show this easily enough for ...
5
votes
2answers
239 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
1
vote
1answer
46 views

For any $c_1,c_2\in\mathbb{C}, E(c_1Z+c_2)=c_1E(Z)+c_2$

When dealing with real-valued RVs, the extensions of expectation and variance are quite clear to me. For example, showing $E(aX+b)=aE(X)+b$ and $var(aX+b)=a^2var(X)$ is relatively straightforward to ...
4
votes
0answers
251 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 ...
1
vote
1answer
185 views

Conditional characteristic function and independence

Suppose we have $$E[\exp{(iu^{\operatorname{tr}}(X_t-X_s))}|\mathcal{G}_s]=\exp{\left(-\frac{1}{2}|u|^2(t-s)\right)}$$ for all $u\in \mathbb{R}^n$. Taking expectation leads to the conclusion that ...
1
vote
1answer
58 views

Deriving the characteristic function for $N(0,2)$

Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
4
votes
3answers
429 views

Is it a characteristic function?

Can anyone explain, how can I prove either $\phi(x) = |\cos t|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
5
votes
1answer
217 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...