Given a finite sequence of positive real numbers $\mathbf w = (w_1,w_2,\ldots,w_n)$, the size biased reordering of $\mathbf w$ is a random vector ${\mathbf w}_o = (w_{r(1)},w_{r(2)},\ldots,w_{r(n)})$ defined by
\begin{equation} \mathbb P(w_{r(1)} = w_j) = \frac{w_j}{\sum_{i=1}^n w_i}, \end{equation} and \begin{equation} \mathbb P(w_{r(k)} = w_j\vert w_{r(1)},w_{r(2)},\ldots,w_{r(k-1)}) = \frac{w_j}{\sum_{i=1}^n w_i - \sum_{i=1}^{k-1}w_{r(i)}}. \end{equation}
My question is the following: Is there an explicit coupling of $w_{r(1)}$ and $w_{r(k)}$ such that \begin{equation} \mathbb P(w_{r(1)}\geq w_{r(k)}) = 1, \end{equation} for a fixed $k$? In other words, is it true that $w_{r(1)}$ stochastically dominates $w_{r(k)}$ for any $k>1$?
edit: the question above is now obsolete.
A follow-up question is: Is it true that \begin{equation} \mathbb E[w_{r(1)}] \geq \mathbb E[w_{r(k)}], \end{equation} for any fixed $k>1$?