Discrete height approximation and weak-cycle estimates

Finite-time reversal identifies forest heights and occupations with weak ascending records and reflected bands. Pre-return duality gives exact mean one at each gap level and almost surely finite weak epochs. Stopped restart gives iid cycles.

The killed-gap Green diagonal equals the weak ladder renewal function. The ladder height tail law, its generating transform and the normalized offspring PGF gap give a scale-uniform second moment, including infinite-variance Gaussian domains.

The maximal inequality and the adjacent-cycle sandwich control the stopped error, with an explicitly reserved extra cycle. Ladder-height exponential tails remove the cutoff. Reversal then gives the two-limit discrete occupation estimate.

Window infima are J1-continuous at paths continuous at their endpoints. Dominated convergence gives occupation continuity outside null boundary levels; these levels fail for at most countably many bands under the stable law. The exact mesh identity has only a final-cell error bounded by 1/(n ε). Continuous mapping therefore gives fixed-band joint walk/occupation convergence.

Choosing bands tending to zero outside the exceptional set and applying the approximation theorem identifies the joint walk/height finite-dimensional limit. At index two, the sealed height identity gives X_t - I_t, with constant one.