Let $X\_1,...,X\_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum\_{i=1}^n w\_i X\_i$ be the weighted sum of these gaussian variables (because $(X\_i)\_{i=1,..,n}$ are not jointly correlated, $S$ can be non normally distributed)


1/ What are the upper bound and/or lower bound of $P(S \leq x)$ $\forall x\in \mathbb{R}$?


2/ And what are the upper bound and/or lower bound of $P(S \leq x)$ if we know the covariance matrix $\Omega$ of these $n$ gaussian random variables?


Could you please recommend me some references on this topic?


Thank you in advance.



Bounds for the sum of dependent gaussian random variables
I found the answer in the theorem 4.9, Bounds for Distribution Functions of Sums of n Random Variables, [chapter 4 of Cherubini, Copula Methods in Finance](https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118673331.ch4) (there is no proof).


Given $S = \sum\_{i=1}^n w\_i X\_i $, the term $P(S \leq s) $ has the lower bound and upper bound as follows


Lower bound:
$$F\_L (s) = \sup\_{\sum\_{i=1}^n t\_i =s} \max \{ \sum\_{i=1}^n F\_{w\_i X\_i}(t\_i)-(n-1) ,0 \}$$


Upper bound:
$$F\_U (s) = \inf\_{\sum\_{i=1}^n t\_i =s} \min \{ \sum\_{i=1}^n F\_{w\_i X\_i}(t\_i) ,1 \}$$


In our particular case, where $X\_i$ follows $N(0,1)$ with $i=1,...,n$, we have


$$F\_L (s) = \sup\_{\sum\_{i=1}^n t\_i =s} \max \{ \sum\_{i=1}^n \Phi(\frac{t\_i}{w\_i})-(n-1) ,0 \}$$
$$F\_U (s) = \inf\_{\sum\_{i=1}^n t\_i =s} \min \{ \sum\_{i=1}^n \Phi(\frac{t\_i}{w\_i}) ,1 \}$$


I don't think we can have close-form expression for these bounds.
$$$$


For the second question, if we know the covariance matrix $\mathbf{\Omega}$. We can make a change of variables as
$$S = \sum\_{i=1}^n w\_i X\_i = \mathbf{w}^T \mathbf{X} = \mathbf{w}^T \sqrt{\mathbf{\Omega} }\mathbf{Z} = \sum\_{i=1}^n w'\_i Z\_i$$
where $Z\_i$ are $n$ gaussian random variables $N(0,1)$ with $i=1,...,n$, $w'\_i$ are the element $i$ of the vector $ \sqrt{\mathbf{\Omega}} \mathbf{w}$, $\sqrt{\mathbf{\Omega}}$ is the square root of the matrix $\mathbf{\Omega}$.


