Suppose we have a random variable $X:\Omega\to[0,1]^d$. Denote partition of $[0,1]^d$ by $\Pi^n = \{\Pi^n (k) \subset [0,1]^d : k = 1,\dots, n\}$ and the corresponding generated sigma-algebra $\sigma^{n} = \sigma(\{X \in \Pi^n(k)\}, k=1,\dots,n)$ where $\sigma(X) = \sigma(\cup_{n\in\mathbb{N}}\sigma^n)$. Suppose $\Pi^{n+1}$ is a refinement of $\Pi^n$ i.e. every component in $\Pi^{n+1}$ is a subset of a component in $\Pi^n$.

For every $w\in\Omega$, define random variable $Y^n$ such that for $k=1,\dots,n$, $$ Y^n(w) = y^n_k \text{ if } X(w) \in \Pi^n(k) $$ where $y^n_k \in \Pi^n(k)$.

My question is whether $Y^n$ converges to $X$ almost surely for any choice of $y^n_k \in \Pi^n(k)$?

Thank you.



You must log in to answer this question.