Conditional-probability inequalities for at-most-k completeness

Determine whether the conditional-probability inequalities relating initial and tail subsequences hold for at-most-k completeness and weak at-most-k completeness of weight sequences generated by independent, identically distributed positive integer-valued gaps; specifically, determine whether conditioning on any finite initial gap sequence yields probabilities at least as large as the corresponding tail-event probabilities.

Background

The paper establishes that asymptotic exactly-k completeness and asymptotic weakly exactly-k completeness are tail events, using the fact that deleting finitely many initial gaps only shifts the resulting weight sequence by a constant. The authors then note that the same argument does not directly extend to at-most-k completeness because it is unclear whether the analogous conditional-probability inequalities remain valid.

The unresolved issue concerns the events denoted by C1≤kC^{\leq k}_1 and C1≤k,wC^{\leq k,w}_1, which represent asymptotic at-most-k completeness and weak asymptotic at-most-k completeness, respectively. Resolving these inequalities would determine whether the corresponding properties can be handled by the same tail-event argument.

References

Note that the same argument does not apply for $\leq k$-completeness as it's not clear why the inequalities $$P(C{\leq k}1 \mid X_1 = x_1, X_2 = x_2, \dots, X_m = x_m) \geq P(C{\leq k}{m+1})$$ $$P(C{\leq k,w}1 \mid X_1 = x_1, X_2 = x_2, \dots, X_m = x_m) \geq P(C{\leq k,w}{m+1})$$ hold.

— Moment conditions for the asymptotic completeness of iid gap sequences  (2503.05177 - Asatryan et al., 7 Mar 2025) in Section 2, immediately before Lemma 2.2 (the lemma labeled \ref{lem:nonprob_leq_equiv_equal})