Stochastic Calculus for the Theta Process

The paper proves that the polygonal lifts X_N = (first level, iterated integrals) of quadratic Weyl sums converge in law in the rough path topology C_g^γ for all γ<1/2 to a geometric rough path above the theta process (Theorem 1.1), by (i) establishing tightness for the level-1 paths X_N, (ii) rewriting the discrete level-2 via a rank-2 theta sum over a triangular cutoff, and (iii) proving tightness and finite-dimensional convergence for the 2-parameter lift using horocycle-lift equidistribution and higher-rank theta technology . By contrast, the model’s tightness argument rests on a Kolmogorov–Chentsov moment criterion, which the paper explicitly shows is insufficient for this problem due to limited moments of the theta process; Kolmogorov yields at best γ<1/4 at level 1, and additional techniques are required to reach all γ<1/2 (Section 6.2, Proposition 6.4) . The paper’s approach carefully implements the needed tail bounds and geometric decompositions; the model’s step (5) is therefore incorrect for this setting.