Basic convergence is a mode of convergence for sequences of functions (or equivalently, signed measures via their distribution functions) introduced by A.A. Khartov (2023) in the context of quasi-infinitely divisible distributions. It was further studied and connected to vague convergence by Stanek (2024).
Let (f_n) and f be functions from R to R. We say (f_n) converges basically to f, written f_n ⇒ f, if every subsequence (f_{n_k}) contains a further subsequence (f_{n_{k_l}}) such that
f_{n_{k_l}}(x) - f_{n_{k_l}}(y) → f(x) - f(y)
for all x, y in R \ S, where S is an at most countable set that may depend on the subsequence.
For signed measures mu_n and mu, basic convergence means that the corresponding distribution functions F^{(mu_n)} converge basically to F^{(mu)}, i.e., every subsequence has a further subsequence along which mu_{n_{k_l}}((x, y]) → mu((x, y]) for all x, y outside a countable exceptional set.
Basic convergence sits in the hierarchy: weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic
Key properties (Stanek 2024):
- Basic convergence is NOT metrizable (Lemma 3.2). The proof uses a Cantor-set-like construction to show that no metric can characterise basic convergence.
- Basic convergence on its own does NOT imply vague convergence. Example 2.7 gives mu_n = n^2 delta_0 - n^2 delta_{1/n} which converges basically to 0 but has unbounded total variation, so does not converge vaguely.
- Combined with local uniform boundedness in variation, basic convergence IS equivalent to vague convergence (Theorem 3.12).
- For non-negative measures, basic convergence is automatically equivalent to vague convergence, since non-negative measures that converge basically are automatically locally bounded in variation (Remark 3.17(c)).
Key Details
- Definition: subsequential convergence of distribution function differences at all but countably many points
- Introduced by: Khartov (2023), for studying quasi-infinitely divisible distributions
- Hierarchy: weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic (all strict for signed measures)
- Main equivalence: vague convergence ⇐> basic convergence + local uniform boundedness in variation (Theorem 3.12)
- Not metrizable: unlike almost basic convergence, basic convergence cannot be metrised
- For non-negative measures: equivalent to vague convergence (the hierarchy collapses)