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)?

Thanks for your help.


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.


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