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In a synchronized network of $n$ nodes, each node will update its parameterbased on the system state in a given iteration. It is well-known that theupdates can converge to a fixed point if the maximum absolute eigenvalue(spectral radius) of the $n \times n$ iterative matrix ${\bf{F}}$ is less thanone (i.e. $\rho({\bf{F}})<1$). However, if only a subset of the nodes updatetheir parameter in an iteration (due to delays or stale feedback) then thiseffectively renders the spectral radius of the iterative matrix as one. Weconsider matrices of unit spectral radii generated from ${\bf{F}}$ due torandom delays in the updates. We show that if each node updates at least oncein every $T$ iterations, then the product of the random matrices (jointspectral radius) corresponding to these iterations is less than one. We thenuse this property to prove convergence of asynchronous iterative systems.Finally, we show that the convergence rate of such a system is$\rho({\bf{F}})\frac{(1-(1-\gamma)^T)^n}{T}$, where assuming ergodicity,$\gamma$ is the lowest bound on the probability that a node will update in anygiven iteration.

Asynchronous Systems and Binary Diagonal Random Matrices: A Proof and Convergence Rate