Epi-Convergence in Distribution

• Epi-convergence in distribution is a mode of weak convergence for random lower semicontinuous functions defined via the convergence of their epigraphs.
• It connects variational convergence, weak convergence of random closed sets in the Fell topology, and asymptotic behavior of ε-optimal solution sets.
• The framework uses measurable selections and local infimum evaluations to analyze the convergence of approximate minimizer sets in stochastic optimization.

Searching arXiv for the cited and closely related papers to ground the article in current literature. arXiv search query: "epi-convergence in distribution normal integrands" Epi-convergence in distribution is a mode of weak convergence for random lower semicontinuous functions, formulated for normal integrands as weak convergence of probability laws on the epi-topological space $(S,\tau_e)$, where $S=S(E)$ denotes the collection of all lower semicontinuous functions $f:E\to\overline{\mathbb R}$. If $Z:\Omega\to(S,\sigma(\tau_e))$ is a normal integrand, then a net $Z_\alpha$ epi-converges in distribution to $Z$ when $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$ converges weakly to $P=\mathbb P\circ Z^{-1}$ on $(S,\tau_e)$; equivalently, under the homeomorphism $\phi:f\mapsto \operatorname{epi}(f)$, the random closed sets $S=S(E)$0 converge in distribution in the Fell topology on $S=S(E)$1 (Ferger, 22 Jul 2025). In this form, the subject connects variational convergence of functions, weak convergence of random closed sets, and asymptotic behavior of $S=S(E)$2-optimal solution sets.

1. Topological and measure-theoretic framework

The 2025 formulation is set on a locally compact, second-countable Hausdorff space $S=S(E)$3, with $S=S(E)$4 the collection of all closed subsets of $S=S(E)$5 and $S=S(E)$6 the collection of all lower semicontinuous functions $S=S(E)$7 (Ferger, 22 Jul 2025). For $S=S(E)$8, the basic set-functionals are

$S=S(E)$9

The Fell topology $f:E\to\overline{\mathbb R}$0 on $f:E\to\overline{\mathbb R}$1 is the smallest topology making $f:E\to\overline{\mathbb R}$2 open for every compact $f:E\to\overline{\mathbb R}$3 and $f:E\to\overline{\mathbb R}$4 open for every open $f:E\to\overline{\mathbb R}$5. The upper-Fell topology $f:E\to\overline{\mathbb R}$6 is generated only by the family $f:E\to\overline{\mathbb R}$7; it is strictly coarser than $f:E\to\overline{\mathbb R}$8 but still Hausdorff and metrizable under the lcscH assumptions (Ferger, 22 Jul 2025).

A random closed set is a measurable map $f:E\to\overline{\mathbb R}$9, where $Z:\Omega\to(S,\sigma(\tau_e))$0. Its distribution is $Z:\Omega\to(S,\sigma(\tau_e))$1, and $Z:\Omega\to(S,\sigma(\tau_e))$2 in $Z:\Omega\to(S,\sigma(\tau_e))$3 means weak convergence of the corresponding laws $Z:\Omega\to(S,\sigma(\tau_e))$4 on $Z:\Omega\to(S,\sigma(\tau_e))$5 (Ferger, 22 Jul 2025).

On the function side, for $Z:\Omega\to(S,\sigma(\tau_e))$6 the epigraph is

$Z:\Omega\to(S,\sigma(\tau_e))$7

which is closed because $Z:\Omega\to(S,\sigma(\tau_e))$8 is lower semicontinuous. The epi-topology $Z:\Omega\to(S,\sigma(\tau_e))$9 is defined as the coarsest topology making the evaluation-infimum maps

$Z_\alpha$0

lower semicontinuous for every compact $Z_\alpha$1 and upper semicontinuous for every open $Z_\alpha$2 (Ferger, 22 Jul 2025). It is explicitly noted that $Z_\alpha$3-convergence is equivalent to classical Attouch-Wets epi-convergence and that

$Z_\alpha$4

is a homeomorphism onto the set $Z_\alpha$5 of all closed epigraphs in $Z_\alpha$6, equipped with the subspace Fell topology (Ferger, 22 Jul 2025).

This formulation excludes a common misunderstanding: epi-convergence in distribution is not pointwise convergence in law of function values, nor is it uniform weak convergence in a supremum metric. The primary object is the random epigraph, and the topology is chosen so that lower semicontinuity and optimization-relevant set limits are preserved.

2. A convergence-determining class for random closed sets

A central technical step is a new characterization of weak convergence on $Z_\alpha$7 (Ferger, 22 Jul 2025). Fix a countable dense subset $Z_\alpha$8 and a countable, dense $Z_\alpha$9 such that every closed $Z$0-ball is a $Z$1-continuity set. Define

$Z$2

Then for probability measures $Z$3 on $Z$4, the following are equivalent: $Z$5 on $Z$6, and for every $Z$7,

$Z$8

Thus weak convergence can be checked on a countable family of hitting-events associated with finite unions of closed $Z$9-balls (Ferger, 22 Jul 2025).

The proof proceeds by constructing a countable base for $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$0 from finite intersections of missing-events and hitting-events, rewriting each basic open as a countable union of sets of the form

$P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$1

with $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$2 and each $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$3 a closed $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$4-ball, and then applying inclusion-exclusion and continuity-set arguments (Ferger, 22 Jul 2025). The result is not merely a convenience: it turns Fell-topology weak convergence into a testable condition involving a countable class of events.

A technical caveat is emphasized in the same source. In general metric spaces one must be careful with interiors of closed balls. If $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$5 is countable with the discrete metric, each closed ball of radius $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$6 is the whole space, whose interior is again $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$7 rather than the single-point open ball familiar from Euclidean spaces. The use of Vaughan’s metric, equivalent to the original topology but making all closed bounded sets compact, is introduced precisely to ensure that closed $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$8-balls behave as compact continuity sets in the Fell topology (Ferger, 22 Jul 2025).

3. Criterion for epi-convergence in distribution of normal integrands

Via the homeomorphism $P_\alpha=\mathbb P_\alpha\circ Z_\alpha^{-1}$9, epi-convergence in distribution of normal integrands reduces to distributional convergence of random closed epigraphs. Applying the convergence-determining class on $P=\mathbb P\circ Z^{-1}$0 with the product metric $P=\mathbb P\circ Z^{-1}$1, the paper derives a necessary and sufficient criterion stated directly in terms of local infima of the random functions (Ferger, 22 Jul 2025).

Let $P=\mathbb P\circ Z^{-1}$2 be the laws of $P=\mathbb P\circ Z^{-1}$3 on $P=\mathbb P\circ Z^{-1}$4, and let $P=\mathbb P\circ Z^{-1}$5 and $P=\mathbb P\circ Z^{-1}$6 on $P=\mathbb P\circ Z^{-1}$7. Then the following are equivalent: first, $P=\mathbb P\circ Z^{-1}$8 in $P=\mathbb P\circ Z^{-1}$9; second, for every finite choice of points $(S,\tau_e)$0, radii $(S,\tau_e)$1, and shifts $(S,\tau_e)$2,

$(S,\tau_e)$3

The statement is equivalent if “$(S,\tau_e)$4” is replaced by “$(S,\tau_e)$5” in the obvious way (Ferger, 22 Jul 2025).

The mechanism is geometric. The event that $(S,\tau_e)$6 hits the closed rectangle $(S,\tau_e)$7 is translated into

$(S,\tau_e)$8

Finite unions of such rectangles then correspond, through inclusion-exclusion, to the ball-based hitting-events needed for Fell-topology convergence (Ferger, 22 Jul 2025). This identifies epi-convergence in distribution as a weak-convergence theory governed by local infimum functionals, not by point evaluations.

4. $(S,\tau_e)$9-optimal solution sets and upper-Fell convergence

For $\phi:f\mapsto \operatorname{epi}(f)$0, the $\phi:f\mapsto \operatorname{epi}(f)$1-optimal solution set is

$\phi:f\mapsto \operatorname{epi}(f)$2

The map $\phi:f\mapsto \operatorname{epi}(f)$3 is jointly measurable from $\phi:f\mapsto \operatorname{epi}(f)$4 into $\phi:f\mapsto \operatorname{epi}(f)$5, and more strongly,

$\phi:f\mapsto \operatorname{epi}(f)$6

is continuous when $\phi:f\mapsto \operatorname{epi}(f)$7 carries the left-order topology $\phi:f\mapsto \operatorname{epi}(f)$8, in which the sets $\phi:f\mapsto \operatorname{epi}(f)$9 are open (Ferger, 22 Jul 2025).

The left-order topology encodes the one-sided condition $S=S(E)$00. Under $S=S(E)$01 in $S=S(E)$02 and $S=S(E)$03 in $S=S(E)$04, the Painlevé-Kuratowski upper bound

$S=S(E)$05

holds (Ferger, 22 Jul 2025). This is the set-valued counterpart of the epi-liminf inequality.

The continuous-mapping theorem then yields the distributional stability of approximate argmin sets. If

$S=S(E)$06

then

$S=S(E)$07

Equivalently, for every finite family of compacts $S=S(E)$08,

$S=S(E)$09

(Ferger, 22 Jul 2025).

The distinction between upper-Fell and Fell convergence is essential. In general, the theory guarantees only upper-Fell convergence of $S=S(E)$10-solution sets. If, however, $S=S(E)$11, the family of exact minimizer sets satisfies the tightness condition

$S=S(E)$12

and $S=S(E)$13 is almost surely a singleton, then the stronger conclusion

$S=S(E)$14

holds (Ferger, 22 Jul 2025). A frequent misconception is therefore that epi-convergence in distribution automatically yields Fell convergence of argmin sets; the general result is upper-Fell, and the Fell conclusion requires additional boundedness and uniqueness assumptions.

5. Measurable selections and convergence to a Choquet-capacity

The theory extends from random closed sets of near-minimizers to measurable selectors. Suppose measurable selections $S=S(E)$15 are chosen almost surely and the family $S=S(E)$16 is tight in the sense that for every $S=S(E)$17 there exists a compact $S=S(E)$18 with

$S=S(E)$19

Then for every closed $S=S(E)$20,

$S=S(E)$21

where $S=S(E)$22 is the capacity-functional of the random set $S=S(E)$23 (Ferger, 22 Jul 2025).

This is a Portmanteau-type weak-convergence statement in which the limit object is a Choquet-capacity rather than, in general, a probability law on $S=S(E)$24. The paper explicitly states that one may regard $S=S(E)$25 as converging in distribution “to the set $S=S(E)$26” in the sense of Portmanteau for capacities (Ferger, 22 Jul 2025). The proof uses upper-Fell convergence of the random closed sets together with a standard Portmanteau-type result for selections from upper-Fell-converging random closed sets due to Ferger.

The probabilistic limit becomes classical when the limiting minimizer is unique. If $S=S(E)$27 and $S=S(E)$28 almost surely for some $S=S(E)$29-valued random variable $S=S(E)$30, then

$S=S(E)$31

(Ferger, 22 Jul 2025). In that case the capacity-functional concentrates on neighborhoods of $S=S(E)$32, and capacity convergence reduces to ordinary weak convergence.

6. Related developments and adjacent uses of epi-type convergence

Several nearby lines of work clarify the scope of epi-convergence in distribution and its relation to other asymptotic frameworks. For expectation functions on metric spaces, sufficient conditions for epi-convergence under varying probability measures and integrands were given in terms of parametric Fatou lemmas, Pasch-Hausdorff envelopes on Suslin metric spaces, and weak-convergence hypotheses combined with uniform integrability from below and above together with pointwise lower and upper semicontinuity conditions (Feinberg et al., 2022). In that setting,

$S=S(E)$33

with applications in sieve estimators, mollifier smoothing, PDE-constrained optimization, and stochastic optimization with expectation constraints (Feinberg et al., 2022). This provides a deterministic expectation-level route to epi-limits when both measures and integrands vary.

A further extension to stochastic dynamic programs uses the Attouch-Wets distance on lower-semicontinuous functions. On $S=S(E)$34,

$S=S(E)$35

and the space is a complete metric space in which closed and bounded sets are compact (Keehan et al., 31 Jan 2025). Under weak convergence of measures, asymptotic semicontinuity and integrability assumptions, and recovery sequences, the expected stage-cost mappings epi-converge; appropriate equi-semicontinuity assumptions then assure epi-consistency of stochastic dynamic programs, including cases with unbounded and simultaneously-approximated stage-cost functions and approximated constraints (Keehan et al., 31 Jan 2025). This suggests that epi-convergence of random or approximate objectives can be propagated through Bellman recursions and fixed-point arguments when the function space is metrized appropriately.

A different but related statistical development appears in weak convergence theory for stochastic processes when uniform weak convergence fails. For locally bounded functions on a locally compact separable metric space, hypi-convergence is defined through simultaneous epi- and hypo-convergence of epigraphs and hypographs, with a semimetric

$S=S(E)$36

after passage to equivalence classes (Bücher et al., 2013). Weak convergence with respect to this metric is weaker than uniform convergence; for continuous limits it is equivalent to locally uniform convergence, and under mild side conditions it implies $S=S(E)$37 convergence (Bücher et al., 2013). Applications include empirical copula processes, tail dependence functions, and residual empirical processes precisely in cases where supremum-norm weak convergence is unavailable.

Taken together, these developments delimit three neighboring uses of epi-type ideas. One concerns random lower semicontinuous functions and their epigraphs as random closed sets (Ferger, 22 Jul 2025). A second concerns epi-convergence of expectation functionals under varying measures and integrands (Feinberg et al., 2022, Keehan et al., 31 Jan 2025). A third concerns weak convergence in semimetric spaces built from epigraphs and hypographs for non-smooth stochastic-process limits (Bücher et al., 2013). The common thread is that epigraph-based topology retains optimization-relevant asymptotics in settings where pointwise or uniform modes of convergence are either too strong or not structurally aligned with the objects of interest.