1
vote
1answer
44 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
1
vote
1answer
57 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
1
vote
0answers
55 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
4
votes
2answers
192 views

Minimizing the expectation over a set, wrt to the Gaussian measure

I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on $\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
1
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1answer
627 views

conditional expectation of normal distribution using sigma algebra

Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities. Let $Y = IX$. How would I find the distribution of $Y$ and ...
5
votes
1answer
206 views

Fractional Part of Sum of Sequence of Independent Normal Random Variables

I'm trying to prove that if $X_n$ iid normal $S_n = \sum_1^n X_i$ $U_n=S_n-\lfloor S_n\rfloor$ then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's ...
0
votes
1answer
264 views

Limit of Sum of Cauchy Random Variables

I'm investigating the behaviour of some random variables obtained from standard Cauchy random variables $X_n$. Suppose $Y_n=\textrm{sgn}(X_n)|X_n|^{\alpha}$ for $\alpha\in[0,1]$. Let ...