Regularity Conditions for Critical Point Convergence

We focus on a sequence of functions ₙ\, defined on a compact manifold with boundary S, converging in the Cᵏ metric to a limit f. A common assumption implicitly made in the empirical sciences is that when such functions represent random processes derived from data, the topological features of fₙ will eventually resemble those of f. In this work, we investigate the validity of this claim under various regularity assumptions, with the goal of finding conditions sufficient for the number of local maxima, minima and saddle of such functions to converge. In the C¹ setting, we do so by employing lesser-known variants of the Poincaré-Hopf and mountain pass theorems, and in the C² setting we pursue an approach inspired by the homotopy-based proof of the Morse Lemma. To aid practical use, we end by reformulating our central theorems in the language of the empirical processes.