Reinforced random walks with geometric inter-transition times

We consider interacting vertex-reinforced random walks on a finite graph, where each walk transitions at independent geometric random times with parameter $p_i \in (0,1]$. The transition matrix of walk $i$ takes the form $Q^i(x, p_i) = p_i \Pi^i(x) + (1-p_i)I$, where $\pi^i(x)$ is the unique invariant measure, independently of $p_i$. Consequently, the limiting points of the occupation measure $X(n)$ coincide with those of the simultaneous-transition model ($p_i = 1$): the solutions of $x = \pi(x)$. Verifying almost sure convergence to these points is non-trivial, since the stochastic input $U(n+1)$ is not a martingale difference. We address this by decomposing $U(n)$ into a convergent martingale, a geometrically decaying component $(1-p)U(n-1)$, and a controlled correction, allowing us to verify the Clark-Kushner condition and establish almost sure convergence.