What is the limit of many processes' minima?

Imagine having several groups of random processes that evolve over time with sudden jumps (cadlag processes). Each group contains many trajectories, and we are interested in the point where each trajectory reaches its lowest value – the so‑called argmin. As the number of groups grows, these minima are no longer fixed numbers but become random variables themselves. The authors show that, as the groups become more numerous, the vector formed by all the argmin converges in distribution to a random closed set. This set is simply the Cartesian product of the minimizer sets of each limiting process, i.e. the processes that appear when the number of groups is sent to infinity. If each limiting process almost surely has a unique minimizer, the random closed set collapses to a single point and the convergence becomes classical: the vector of argmin tends to a deterministic vector made of the minimizers of the limiting processes. The result links the theory of convergence of closed sets with the classical argmin theory, providing a tool to study optimization procedures that depend on many noisy data, such as those found in machine learning or quantitative finance.

The extended argmin theorem deals with a family of nets (directed indices) {X_i^α}_{α∈A_i} of multidimensional stochastic processes with cadlag trajectories, where each index i labels a different component. For each net we define the argmin set M_i^α = {x ∈ ℝ^d : X_i^α(x) = inf_t X_i^α(t)}. Assuming the nets converge in the Skorokhod J1 topology to limit processes X_i, the main result states that the random closed vector (M_1^α,…,M_k^α) converges in distribution, under the Fell topology, to the random closed set M = M_1 × … × M_k, where each M_i is the argmin set of the limit process X_i. The proof relies on the continuity of the argmin operator with respect to uniform convergence on compacts and the closedness of the minimizer set under small perturbations, together with the representation theorem for random closed sets via support functionals. When each M_i is almost surely a singleton, the Cartesian product collapses to a deterministic point and Fell convergence reduces to ordinary convergence of random vectors, thus recovering the classical argmin theorem. This framework enables the analysis of situations where the input data are themselves stochastic processes, e.g., estimation procedures based on time‑series samples or stochastic optimization algorithms where the objective function is observed with noise.