We develop a general asymptotic theory for estimating equations whose dimension diverges with the sample size. For both unpenalized and sparse penalized problems, we establish population-level conditions for existence, consistency, uniqueness, and asymptotic normality; in the penalized setting, we also establish selection consistency under a generalized version of the mutual incoherence condition. Our results cover stepwise procedures with a diverging number of steps, independent but non-identically distributed observations, and dependent data. We allow penalties that are simultaneously nonconvex, non-coordinate-separable, and group-structured and may involve heterogeneous tuning parameters. Our population-level conditions imply a weak form of restricted strong convexity, and we provide an explicit example where the commonly used stronger form fails. The results are illustrated by several applications including Group SCAD-penalized estimation in generalized linear models, distributed inference under network dependence, and penalized stepwise estimation in causal inference.

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