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The design of online algorithms for matching markets and revenue management settings is usually bound by the stochastic prior that the demand process is formed by a fixed-length sequence of queries with unknown types, each drawn independently. This assumption of {\em serial independence} implies that the demand of each type, i.e., the number of queries of a given type, has low variance and is approximately Poisson-distributed. This paper explores more general stochastic models for online edge-weighted matching that depart from the serial independence assumption. We propose two new models, \Indep and \Correl, that capture different forms of serial correlations by combining a nonparametric distribution for the demand with standard assumptions on the arrival patterns -- adversarial or random order. The \Indep model has arbitrary marginal distributions for the demands but assumes cross-sectional independence for the customer types, whereas the \Correl model captures common shocks across customer types. We demonstrate that fluid relaxations, which rely solely on expected demand information, have arbitrarily bad performance guarantees. In contrast, we develop new algorithms that essentially achieve optimal constant-factor performance guarantees in each model. Our mathematical analysis includes tighter linear programming relaxations that leverage distribution knowledge, and a new lossless randomized rounding scheme in the case of $\Indep$. In numerical simulations of the $\Indep$ model, we find that tighter relaxations are beneficial under high-variance demand and that our demand-aware rounding scheme can outperform stockout-aware rounding.