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

A Class of Doubly Stochastic Shift Operators for Random Graph Signals and their Boundedness