In the proof of a theorem my lecturer seemed to have used this fact without first proving it: Let $(X_i)_{i \geq 1}$ be real-valued independent random variables on $(\Omega,\mathscr{F},\mathbb{P})$, and let $\mathbb{N}$ be partitioned into disjoint countable sets $(I_k)_{k\geq 1}$. If $Y_k:=\sum_{i \in I_k} X_i$ converges for all $k$, then $(Y_k)_{k\geq 1}$ are independent random variables. (It is clear to me that each $Y_k$ is a random variable, but I need to prove independence.)

Specifically, the proof uses this fact with $X_i = $ the $i$-th Rademacher function, suitably scaled.

Can somebody give a hint please?

Edit Independence of random variables is defined in terms of the corresponding $\sigma$-algebras on $\Omega$. For real-valued random variables, I understand that $(Y_k)_{k\geq1}$ are independent if and only if $\mathbb{P}(Y_1 \leq y_1, \dotsc, Y_k \leq y_k) = \prod_{1\leq j \leq k}\mathbb{P}(Y_j \leq y_j)$ for all $k$.

  • $\begingroup$ How you have defined independence for a sequence of random variables $\{X_i\}_{i=1}^{\infty}$ in your class at this point? $\endgroup$ – Tom Nov 7 '13 at 18:03
  • $\begingroup$ @Tom Please see the edit! $\endgroup$ – user71815 Nov 7 '13 at 18:11

We have to show that for each integer $N$, $(Y_j,1\leqslant j\leqslant N)$ are independent random variables. So take $B_j,1\leqslant j\leqslant N$ some Borel sets. We have $\{Y_j\in B_j\}\in \sigma(X_i,i\in I_j)=:\mathcal F_j$, so we have to show that the $\sigma$-algebras $(\mathcal F_j,1\leqslant j\leqslant N)$ are independent.

By an approximation argument, it's enough to see that $\left(\bigcup_{l=1}^\infty\mathcal F_j^{(l)},1\leqslant j\leqslant N\right)$ are independent, where $\mathcal F_j^{(l)}:=\sigma(X_i,i\in I_j\cap [-l,l])$. Indeed, for each $j$, $\mathcal F_j$ is generated by the algebra $\bigcup_{l=1}^\infty\mathcal F_j^{(l)}$.

  • $\begingroup$ Can you elaborate on "$I_j \cap [-l,l]$" please? $I_j$ is a subset of $\mathbb{N}$, so $\mathcal{F}_j^{(1)} \subseteq \mathcal{F}_j^{(2)} \subseteq \dotsb$, giving $\bigcup_{l=1}^\infty \mathcal{F}_j^{(l)} = \mathcal{F}_j$? Do you mean we should show $(\mathcal{F}_j^{(l)}, 1\leq j \leq N, l \in \mathbb{N})$ are independent? $\endgroup$ – user71815 Nov 8 '13 at 15:52
  • $\begingroup$ The union is not necessarily $\cal F_j$, but it is an algebra which generates this $\sigma$-algebra. For your second question, yes. $\endgroup$ – Davide Giraudo Nov 8 '13 at 15:58
  • $\begingroup$ Okay, would you mind editing your answer? $\endgroup$ – user71815 Nov 8 '13 at 16:00
  • $\begingroup$ I made a small edit. Do you need more details? $\endgroup$ – Davide Giraudo Nov 8 '13 at 21:05
  • $\begingroup$ You have basically reduced the question to "If $X_1,X_2,X_3$ are independent random variables, then $\sigma(X_1,X_2)$ and $\sigma(X_3)$ are independent", am I right? I'll need to find a proof of this... $\endgroup$ – user71815 Nov 8 '13 at 22:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.