I am trying to get a head start on the next semester at uni. The following question is based on the statistical problem and solution outlined on pages 3 to 5 of this book.

The problem is based on computing the expected number of returns to the origin $$\left(N\right)$$ of a random walk.

My question is around interpreting the notation the author has used, in particular:

1. $$E\left(N\right) = \sum_{n\geq1} \left(n\right)P\left(N=n\right) =\sum_{n\geq1}\sum_{m=1}^{n}P\left(N=n\right)$$

I read the summation index $$n\geq1$$ as if I can start at any value of $$n$$ that is greater than one and sum to infinity such as $$\sum_{n=5}^{\infty}…$$. However, because this is an expectation, it doesn't make sense that you can start at any value of $$n$$ as there is a positive expectation for $$n=2, n=4,... etc$$.

Within this context, would I be correct in interpreting the first summation as the sum from $$n=1$$ to $$\infty$$, such as:

1. $$E\left(N\right) = \sum_{m=1}^{1}P\left(N=1\right) + \sum_{m=1}^{2}P\left(N=2\right) + \sum_{m=1}^{3}P\left(N=3\right) + …$$

Furthermore, the text goes on to say:

1. $$E\left(N\right) = \sum_{n\geq1}\sum_{m=1}^{n}P\left(N=n\right) = \sum_{1 \leq m \leq n}P\left(N=n\right)$$.

If my interpretation for (2) is correct, how are you meant to read $$\sum_{1 \leq m \leq n}P\left(N=n\right)$$ to recover (2)?

The symbol $$\sum_{n\geq 1} a_n$$ indicates the sum of the elements of the sequence $$a_n$$ for all $$n$$ that are greater than or equal to 1, so in this case it would be $$\sum_{n\geq 1} n P(N=n) = 1 P(N=1) + 2 P(N=2) + 3 P(N=3) + \cdots$$ which is consistent with your interpretation 2.
The symbol $$\sum_{1 \leq m \leq n} a_n$$ that is in your last point is to be read "for all $$n$$ and $$m$$ that satisfy $$1 \leq m \leq n$$"; so again it is consistent with 2.