Definition:Independent Random Variables/General Definition

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

For each $i \in \N$, let $\map \sigma {X_i}$ be the $\sigma$-algebra generated by $X_i$.

We say that $\sequence {X_n}_{n \mathop \in \N}$ is a sequence of independent random variable :


 * $\sequence {\map \sigma {X_n} }_{n \mathop \in \N}$ is a sequence of independent $\sigma$-algebras.