Independence of Events and of Random Elements

In the setting of Probability Spaces: Notation, let $J$ be a set.

(i) Events. A family $(A_{j})_{j\in J}$ of events is independent if the family $(\{A_{j}\})_{j\in J}$ of subsets of $\mathcal{F}$ is independent.

(ii) Random elements. Let $((S_{j},d_{j}))_{j\in J}$ be a family of metric spaces and let $(X_{j})_{j\in J}$ be a family in which each $X_{j}$ is a random element of $S_{j}$. The family $(X_{j})_{j\in J}$ is independent if the family $(\sigma(X_{j}))_{j\in J}$ of the $\sigma$-algebras generated by the $X_{j}$ is independent.

(iii) Two random elements. Random elements $X$ and $Y$ of metric spaces $S$ and $T$ are independent if the $\sigma$-algebras $\sigma(X)$ and $\sigma(Y)$ are independent.

(iv) A random element and a $\sigma$-algebra. A random element $X$ of a metric space $S$ is independent of a $\sigma$-algebra $\mathcal{G}$ on $\Omega$ contained in $\mathcal{F}$ if $\sigma(X)$ and $\mathcal{G}$ are independent.

(v) Independent and identically distributed sequences. Let $(S,d)$ be a metric space. A sequence $(X_{k})_{k\in\mathbb{N}}$ of random elements of $S$ is independent and identically distributed (iid) if it is independent, as in (ii), and $X_{k}$ and $X_{l}$ are identically distributed for all $k,l\in\mathbb{N}$.