Let \begin{equation} P= \{(x_i)_{i\in\mathbb{N}}: x_i\geq 0, \sum_{i=1}^\infty x_i=1\} \end{equation} Every $p\in P$ is called probability vector.
The following notation is introduced in a book, but they, $P_m$ and $p_{(m)}$, are not clear for me. Can you help me with an example to understand them?
Assume that $P_m$ denote the subset of $P$ consisting of all $m$-dimensional probability vectors i.e. satisfying $p_i=0$ for all $i>m$.
For $p\in P$, we let $p_{(m)}\in P_m$ denote the vector obtained from $p$ by taking its $m-1$ largest term and, as the $m$th term, the sum of the rest, and ordering the resulting $m$-terms decreasingly.