Dot Product of Random Vector and Independent Entry Random Vector

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{M}, \mathbb{P})$ be a Probability Space. Suppose $(X_n)$ is a Sequence of Random Variables and $(Y_{n})$ is a Sequence of Independent Random Variables that is also independent with $X_{n}$ for all $n \in \mathbb{N}$. Denote $$\begin{align*} &\mathbb{E}(X_{n}) = \mu_{n} & \mathbb{E}(Y_{n}) = \nu_{n}\\ &\text{Var}(X_{n})&a\\ &S_{n} = \sum\limits_{n=1}^{\infty} X_{n} Y_{n}\\ \end{align*}$$

TODO

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