I'm having these definitions in the proof for the law of large numbers which I'm trying to understand in detail:

\begin{align*} k &> 1 \\ J_1 &= \mathbf{1}_{[0 \lt \mid X_1 \mid \le 1]} \\ J_k &= \mathbf{1}_{[k-1 \lt \mid X_1 \mid \le k]} \\ I_n &= \sum_{k=1}^n J_k \end{align*}

Where $\mathbf{1}_A$ is the indicator function defined as

\begin{align*} \mathbf{1}_A(\omega) := \begin{cases}1 \text{, if } \omega \in A \\ 0 \text{, if } \omega \notin A\end{cases} \end{align*}

and $X_1$ is a random variable.

My problem here is that I cannot really understand what $\mathbf{1}_{[k-1 \lt \mid X_1 \mid \le k]} \,$ does mean here. $[k-1 \lt \mid X_1 \mid \le k]$ would be the set of all values which are larger than $k-1$ and less or equal than $k$, I get that. But since $X_1$ is not closer defined I fail to see what this set looks like and that leads to the point where I do not understand the sum $I_n$.

Could somebody explain to me what I am actually looking at? Please let me know if you need more information and please note that I do not have a mathematical background in particular.

  • $\begingroup$ $$I_n= \mathbf{1}_{[0 \lt \mid X_1 \mid \le n]}$$ $\endgroup$ – Did Oct 30 '16 at 12:08
  • $\begingroup$ @Did I think I don't understand this. $\mathbf{1}_X$ is a function on a set $X$. How can I understand the notation of summing over such a function? $\endgroup$ – displayname Oct 30 '16 at 13:37
  • $\begingroup$ Simply note that $[0 \lt|X_1|\le 1]$ is a shortcut for $\{\omega\in\Omega\,;\,0 \lt|X_1(\omega)|\le 1\}$. $\endgroup$ – Did Oct 30 '16 at 14:31

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