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**cribby** I think I miscommunicated my difficulty with this part of the proof. What would help me get up to speed here is a genetic account of the nonzero portion of the limit r.v. I get that everything has to go somewhere and so we send the stuff we're really not interested in to some garbage value, but I'm not clear why the nonzero portion was defined the way it was. I guess what I'm looking for is some heuristic enlightenment--how, by what brain crevasses, would one generate this?

Oh, I see... :-s

There is a very good reason for this definition: in wide generality, for a sequence $\displaystyle (a_n)_n$, we have $\displaystyle a_n=a_1+(a_2-a_1)+\cdots+(a_n-a_{n-1})$ ("telescoping sum": all terms but one cancel), hence

($\displaystyle (a_n)_n$ converges toward $\displaystyle a$) iff (the series $\displaystyle \sum_k (a_{k+1}-a_k)$ converges and $\displaystyle a_1+\sum_{k=1}^\infty (a_{k+1}-a_k)=a$).

The series is just a convenient way to rewrite the limit since we can use one of the numerous convergence tests (here, the comparison test). Another way of writing the proof would be: "(....) thus almost-surely the series $\displaystyle \sum_k (X_{n_k}-X_{n_{k-1}})$ converges (since it converges absolutely), which is to say that almost-surely the sequence $\displaystyle (X_{n_k})_k$ converge (since $\displaystyle X_{n_k}=X_{n_1}+\sum_{i=2}^k (X_{n_i}-X_{n_{i-1}})$ for all $\displaystyle k$). Let us define $\displaystyle X$ to be the limit of this sequence when it converges, and 0 otherwise. "

I found another possible correction to the statement in the course notes--that "fundamental

*almost surely*" is equivalent to convergence almost surely.

That indeed would be correct since it is just another way of saying that a sequence converges if (and only if) it is Cauchy.