3
votes
Accepted
Given a random vector has univariate normal marginals, and a positive definite Covariance. Does this mean the vector is multivariate normal?
Let $X$ be a standard one-dimensional normal variable, and let $J$ be independent of $X$ with $$P(J=1)=2/3=1-P(J=-1)\,.$$ Define
$Y=XJ$. Then $X,Y$ are both $N(0,1)$ variables, and $$E(XY)=E(X^2)E(J)=...
- 15.9k
2
votes
How sub-Gaussian is a Truncated Normal?
I know it has been a while since you posted this but just in case you are still interested in the question, here is what I had found while working on a similar problem.
Let $X$ be a sub-Gaussian ...
- 1,314
2
votes
Accepted
Does closeness in total variation distance imply closeness in moments?
No, the total variation norm does not imply that moments are close. Let $f$ be the density of $X$ and let $g$ be any other probability density function. Let $Y$ be the random variable with pdf $(1-\...
- 8,268
1
vote
Accepted
Sum of Bernoulli random variables with random coefficients
Consider $S\equiv \{(b_k) \in \mathbb R^{\infty}: \frac 1 n \sum\limits_{k=1}^{n} b_k^{2}\to 1\}$.
By the ordinary CLT we get $\frac 1 {\sqrt n} \sum\limits_{k=1}^{n} b_kx_k \to N(0,1)$in ...
- 5,658
1
vote
The root of the sum of two normally distributed variables
It is impossible to say anything about the distribution of $Z$ without extra hypothesis. I will assume that $X$ and $Y$ are independent so that the question does have an answer.
In this case we have $...
- 5,658
1
vote
The root of the sum of two normally distributed variables
Perhaps your approach is fruitful?
$$
F_Z(z)
= \mathbb{P}\left[\sqrt{X^2+Y^2} < z\right]
= \iint_{D_z} f_X(x) f_Y(y) dxdy,
$$
where $D_z$ is the region bounded by $\sqrt{x^2+y^2}<z$.
Since $X,...
- 52.6k
Only top scored, non community-wiki answers of a minimum length are eligible