Edgeworth expansion for m‑dependent integer‑valued variables

Develop an Edgeworth expansion theorem for sums of m‑dependent integer‑valued random variables (lattice case), including appropriate discrete correction terms and uniform error bounds, analogous to existing non‑lattice results for m‑dependent sequences.

Background

In the application to Litt’s game, the score increments form an m‑dependent sequence. The literature contains Edgeworth expansions for m‑dependent variables in the non‑lattice case, but the authors explicitly state they have not found a suitable theorem for the lattice (integer‑valued) case.

Establishing such a result would directly complement the tools used in the paper and broaden applicability to discrete dependent structures.

References

An alternative approach would be to use results on Edgeworth expansions for sums of m-dependent variables. However, we have not found a suitable such theorem for integer-valued variables; for the non-lattice case, see e.g. Heinrich [11], Rhee [24], and [18]; see also Rinott and Rotar [25] for a more general result, and the further references there.

— The generalized Alice HH vs Bob HT problem  (2503.19035 - Janson et al., 24 Mar 2025) in Section 2.3 (Background on Edgeworth expansions)

Conjecture 1 (mean drift; formal expansion, remainder not bounded). Under (A1)–(A3), writing ρ′I = dρI /dp,E[C] − p = A(p)n + Rn, A(p) = m − 2np(1 − p) ρ′I (p) − (2p − 1) ρI (p)o,with Rn not bounded here. Equivalently, and equally conjecturally, b * (E[C] − p) → m−1 2m {p(1 −p)ρ′I (p) − (2p − 1)ρI (p)}.

— The Exceedance Design Effect: Effective Sample Size for Thresholds under Clustering  (2608.21262 - Noonan, 21 Aug 2026) in Conjecture 1 and Section 10, pp. 8, 34, Appendix B.7, p. 42