This paper seeks to address two neglected aspects of convergence dynamics of cross-country per capita income. First, we allow evolutionary path of per capita income to contain stochastic shocks which may not converge fast enough to the long-run mean. Under this condition, we show that the conventional inference on σ convergence can be enlarged with more predictive power if one assumes, along with the necessary condition of β convergence, that the stochastic shocks are covariance stationary. Second, we argue that for economies to (conditionally) converge, they need to be sufficiently cohesive so that the growth of stochastic shocks is not sustained through complex socio-economic interactions. Empirical examination is carried out by analyzing time series properties of state per capita income in India and performing convergence analysis by conditioning a constructed social cohesion index based on indicators collected from the National Sample Survey. It is demonstrated that when the economy faces monotonic social segmentation, persistence of stochastic shocks considerably affects speed of per capita output convergence.