Definition:Independent Random Variables/General Definition
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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 if and only if:
- $\sequence {\map \sigma {X_n} }_{n \mathop \in \N}$ is a sequence of independent $\sigma$-algebras.
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $4.1$