3 votes
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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)=...
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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 ...
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2 votes
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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-\...
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1 vote
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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 ...
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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 $...
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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,...
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