The concept of relative convergence, which requires the ratio of two time series to converge to unity in the long run, explains convergent behavior when series share commonly divergent stochastic or deterministic trend components. Relative convergence of this type does not necessarily hold when series share common time decay patterns measured by evaporating rather than divergent trend behavior. To capture convergent behavior in panel data that do not involve stochastic or divergent deterministic trends, we introduce the notion of weak σ-convergence, whereby cross section variation in the panel decreases over time. The paper formalizes this concept and proposes a simple-to-implement linear trend regression test of the null of no σ-convergence. Asymptotic properties for the test are developed under general regularity conditions and various data generating processes. Simulations show that the test has good size control and discriminatory power. The method is applied to examine whether the idiosyncratic components of 46 disaggregate personal consumption expenditure (PCE) price inflation items σ-converge over time, finding strong evidence of weak σ-convergence in these data. In a second application, the method is used to test whether experimental data in ultimatum games converge over successive rounds, again finding evidence in favor of weak σ-convergence. A third application studies convergence and divergence in US States unemployment data over the period 2001–2016.