What does it mean if a sequence of random variables $(X_i)_{i\in\mathbb N}$ is independent of a random variable $N$?
Does it mean that for every $n\in\mathbb N$:
$$P(X_1\le x_1,\dots,X_n\le x_n, N\le n)=P(X_1\le x_1,\dots,X_n\le x_n)\cdot P(N\le n)?$$
Best Answer
In general, it is easier to define the following: Two $\sigma$-algebras $\mathcal{F}$ and $\mathcal{G}$ are said to be independent under a probability measure $\mathbb{P}$ if $\mathbb{P}(F\cap G)=\mathbb{P}(F)\mathbb{P}(G)$ for all $F\in \mathcal{F}$ and $G\in \mathcal{G}$.
We then define two measurable maps $Y$ and $Z$ to be independent if $\sigma(Y)$ is independent of $\sigma(Z)$. This, in particular, implies in the case $Y=(X_n)_{n\in \mathbb{N}}$ and $Z=N$ that $\mathbb{P}((X_{n_1}\in B_1,...,X_{n_k}\in B_k)\cap (N\in B_{k+1}))=\mathbb{P}((X_{n_1}\in B_1,...,X_{n_k}\in B_k))\mathbb{P}(N\in B_{k+1})$ for all finite collections of Borel sets $(B_j)_{1\leq j\leq k+1}$ and $n_1<n_2<...<n_k$.