4.1 Operator and convergence certificate
A leave-one-out simplex operator is nonnegative, row stochastic, and zero-diagonal. Its induced infinity norm is at most one, so \(\lVert \rho W\rVert _\infty {\lt}1\) whenever \(|\rho |{\lt}1\).
The simplex constraints give the row sums and nonnegativity; the norm gate follows from the maximum absolute row sum.
For \(P=\mathbf1\pi ^\top \), a geometric Perron certificate records constants \(C\geq 0\) and \(0\leq q{\lt}1\) such that \(\lVert W^k-P\rVert \leq Cq^k\) for every natural number \(k\).
A geometric Perron certificate implies \(W^k\to P\).
The geometric upper bound converges to zero because \(q{\lt}1\).
Under the Perron limit, \(W^kx\to (\pi ^\top x)\mathbf1\) for every vector \(x\). The limit is a constant vector, not a traded basket.
Apply the matrix limit to \(x\).
Under the Perron limit, the limiting vector obeys \(\pi ^\top W=\pi ^\top \).
Pass the limit through one additional multiplication by \(W\).