Kolmogorov's 3-series theorem states that if $X_k, k \geq 1$ are independent, $A > 0$ a threshold level, $Y_k = X_k 1(|X_k|\leq A)$,
then $\sum_k X_k$ converges iff all three series converge:
- $\sum_{k \geq 1} P[|X_k| > A] < \infty$
- $\sum_{k \geq 1} E[Y_k]$ converges
- $\sum_{k \geq 1} Var[Y_k] < \infty$
In the lecture it was mentioned that 1. and 3. are absolute convergence, whereas 2. is convergence of partial sums.
First of all: is this clear from notation? So if we write $< \infty$, then it is not clear that the sum converges? But since the terms all are non-negative, we have that the sequence must converge, and hence we also have absolute convergence? When it is written that a serie converges, then this means that it does not oscillate but goes to a fixed value $(< \infty)$? But it could be that some of the terms cancel, so that we would not have absolute convergence?
Why do we write $< \infty$ for absolute convergence and "converges" for convergence of partial sums? Or has this something to do with the nonnegativity? Would it be wrong to write in 2. $\sum_{k \geq 1} E[Y_k] < \infty$, because that could also imply that the series jumps between some values that are finite?
When can we do reordering of the terms? Whenever they converge or only with absolute convergence?
Sorry for the weird questions, but I do not study maths and only attended a very basic analysis and probability course. Thanks a lot!