Sum independent normal variables

[FactChecker]

Covariance $E(\xi\eta)=E(\sigma\xi^2)=E(\sigma)E(\xi^2)=0$, since $E(\sigma)=0$ while $\sigma$ is independent of $\xi$ and both means $=0$.

So this is a good example to show that even uncorrelated is not enough to guarantee that the sum is normal. It requires independence.

 

[mathman]

So this is a good example to show that even uncorrelated is not enough to guarantee that the sum is normal. It requires independence.

Independence is not necessary If the variables are jointly normal, even if correlated, the sum will be normal.

 

[FactChecker]

Independence is not necessary If the variables are jointly normal, even if correlated, the sum will be normal.

I stand corrected. I should have just said that uncorrelated is not enough.

 

[WWGD]

Are their theorems that deal with the general question? - Given a set of random variables $\{X_1,X_2,..,X_M\}$, when does their exist some subspace $S$ of the vector space of random variables such that $S$ contains each $X_i$ and $S$ has a basis of mutually independent random variables?
"Linearly independent" presumably implies we are considering a set of random variables to be a vector space under the operations of multiplication by scalars and addition of the random variables.
Suppose we take "transformed" to mean transformed by a linear transformation in a vector space. If the vector space containing the random variables ${X_1,X_2,...X_M}$ has a finite basis $B_1,B_2,...B_n$ consisting of mutually independent random variables then (trivially) for each $X_k$ there exists a possibly non-invertible linear transformation $T$ that transforms some linear combination of the $B_i$ into $X_k$.

If the smallest vector space containing the $X_i$ is infinite dimensional (e.g the vector space of all measurable functions on the real number line) , I don't know what happens.

I don't recall any texts that focus on vector spaces of random variables. Since the product of random variables is also a random variable, the topic for textbooks seems to be the algebra of random variables. But that approach downplays the concept of probability distributions.

Would be interesting to see if there is something similar to "Decouple" a basis for such space, so that any two $X_i, X_j$ are independent but the span is preserved.

 

[FactChecker]

Would be interesting to see if there is something similar to "Decouple" a basis for such space, so that any two $X_i, X_j$ are independent but the span is preserved.

That would require a change of the independent variables to a different set.