  A class of doubly stochastic graph shift operators (GSO) is proposed, which
is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally
stationary random graph signals; (ii) $L_{2}$-isometry for \textit{i.i.d.}
random graph signals with the asymptotic increase in the incoming neighbourhood
size of vertices; and (iii) preservation of the mean of any graph signal. These
properties are obtained through a statistical consistency analysis of the graph
shift, and by exploiting the dual role of the doubly stochastic GSO as a Markov
(diffusion) matrix and as an unbiased expectation operator. Practical utility
of the class of doubly stochastic GSOs is demonstrated in a real-world
multi-sensor signal filtering setting.
