Exponent persistence under temporal dependence

Determine conditions on a dependent symbol sequence with marginal distribution π under which the exponent 2 and the orientation ratio 2^γ persist for the limiting directed and undirected degree distributions of the range-renewal graph.

Background

The results concern independent and identically distributed symbols with a regularly varying sampling law π. The proof relies on infinite-occupancy estimates for discovery times and residual unseen mass, together with geometric representations of inter-discovery gaps.

For dependent symbol sequences having the same marginal law π, temporal dependence can alter both discovery gaps and repeated transitions. The paper leaves unresolved which dependence assumptions preserve the universal degree exponent 2 and the factor 2γ relating undirected and directed asymptotics.

References

Second, for a dependent symbol sequence with marginal law $\pi$, temporal dependence changes both discovery gaps and transition repetitions. Determining conditions under which the exponent $2$ and the orientation ratio $2\gamma$ persist would connect the present occupancy argument to sequence-generated graphs from correlated data.

— Universal Exponent-Two Degree Laws in Range-Renewal Networks  (2609.05290 - Xie et al., 4 Sep 2026) in Section 6, Discussion and outlook