  The popular $\mathcal{AB}$/push-pull method for distributed optimization
problem may unify much of the existing decentralized first-order methods based
on gradient tracking technique. More recently, the stochastic gradient variant
of $\mathcal{AB}$/Push-Pull method ($\mathcal{S}$-$\mathcal{AB}$) has been
proposed, which achieves the linear rate of converging to a neighborhood of the
global minimizer when the step-size is constant. This paper is devoted to the
asymptotic properties of $\mathcal{S}$-$\mathcal{AB}$ with diminishing
stepsize. Specifically, under the condition that each local objective is smooth
and the global objective is strongly-convex, we first present the boundedness
of the iterates of $\mathcal{S}$-$\mathcal{AB}$ and then show that the iterates
converge to the global minimizer with the rate
$\mathcal{O}\left(1/\sqrt{k}\right)$. Furthermore, the asymptotic normality of
Polyak-Ruppert averaged $\mathcal{S}$-$\mathcal{AB}$ is obtained and
applications on statistical inference are discussed. Finally, numerical tests
are conducted to demonstrate the theoretic results.
