2
votes
1answer
106 views

Characteristic function of an integer-valued distribution, inversion formula

I am working on the following: Show that if $\varphi$ the characteristic function of an integer-valued distribution then \begin{align*} \mathbb P(X = k) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-itk} ...
0
votes
0answers
40 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
2
votes
2answers
87 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} ...
0
votes
1answer
48 views

Question on proof of Schoenberg correspondence from Lévy Process and Stochastic Calculus by Applebaum

I quote the proof here from Applebaum's Lévy Processes and stochastic calculus (and the things before it to present the full picture) We say $\phi:\mathbb{R}^d\rightarrow\mathbb{C}$ is conditionally ...
1
vote
1answer
48 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 ...