What is the limit of this (complicated) set?

[Adeimantus]

This is going to take a while to set up, so I apologize for that. This came up in the course of thinking about the Strong Law of Large Numbers. It's not homework.

Suppose you have a doubly infinite sequence of random variables $X_{i,n}$ that obey the following almost sure convergence relations. For each $i = 1,2,3,...$,

$$X_{i,n} \xrightarrow{a.s} a_i \quad \mbox{ as } \quad n\xrightarrow{} \infty$$.

Further, we have that $\sum_{i=1}^\infty a_i = \mu < \infty$. Since this series converges, for any $\delta > 0$, there is some smallest $I$ such that $\left| \sum_{i=1}^m a_i - \mu \right | < \delta$ for all $m \geq I$. Consider a sequence of deltas decreasing to zero, and the increasing sequence of their corresponding $I$'s.

$$\delta_1 > \delta_2 > ... \xrightarrow{} 0 \quad \mbox{and} \quad I_1 < I_2 < ...$$

Consider some particular pair $(\delta_j, I_j)$. Since almost sure convergence is linear,

$$\sum_{i=1}^{I_j}X_{i,n} \xrightarrow{a.s} \sum_{i=1}^{I_j} a_i \quad \mbox{ as } \quad n\xrightarrow{} \infty$$

This is the same thing as saying the set

$$\{ \omega: \left| \sum_{i=1}^{I_j}X_{i,n}(\omega) - \sum_{i=1}^{I_j} a_i \right| > \epsilon \quad i.o. \quad n\xrightarrow{} \infty \}$$

has probability zero for any choice of $\epsilon > 0$. From the definition of the deltas and I's, the set

$$A_j = \{ \omega: \left| \sum_{i=1}^{I_j}X_{i,n}(\omega) - \mu \right| > \epsilon + \delta_j \quad i.o. \quad n\xrightarrow{} \infty \}$$

also probability zero. My question is, does the sequence of sets $A_j$ have a limit of

$$A = \{ \omega: \left| \sum_{i=1}^{\infty}X_{i,n}(\omega) - \mu \right| > \epsilon \quad i.o. \quad n\xrightarrow{} \infty \}$$

Thanks for wading through that!

 

[andrewkirk]

I waded some of the way, then got stuck here:

$$\left\{ \omega: \left| \sum_{i=1}^{I_j}X_{i,n}(\omega) - \sum_{i=1}^{I_j} a_i \right| > \epsilon \quad i.o. \quad n\xrightarrow{} \infty \right\}$$

I have not come across the initials $i.o.$ before. What do they mean?

I feel that perhaps the set is
$$\left\{ \omega:<br /> \forall M\in\mathbb N\ \exists n\ge M\ :\<br /> \left| \sum_{i=1}^{I_j}X_{i,n}(\omega) - \sum_{i=1}^{I_j} a_i \right| > \epsilon \right\}$$
in which case the statement is that the set of $\omega$ for which the $n$-indexed sequence of sums of the first $I_j$ RVs does not converge to the sum of the first $I_j$ $a_i$s, has probability measure zero.

Is that what you meant?