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).
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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 ...
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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 ...
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0answers
452 views
Characteristic functions based proof problem.
I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
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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 ...
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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.
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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.
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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, ...
