2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229659We consider the ensemble of curves $\{γ_{α,N}:α\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(πi n^2α)$, $0\leq N'<N$. We prove the existence of limiting finite-dimensional distributions for such curves as $N\to\infty$, with respect to an absolutely continuous probability measure $μ_R$ on $(0,1]$. Our Main Theorem generalizes a result by Marklof and Jurkat and van Horne. Our proof relies on the analysis of the geometric structure of such curves, which exhibit spiral-like patterns (curlicues) at different scales. We exploit a renormalization procedure constructed by means of the continued fraction expansion of $α$ with even partial quotients and a renewal-type limit theorem for the denominators of such continued fraction expansions.36 pages, 3 figures, submitted to Ann. Inst. Henri Poincare' Probab. StatDynamical SystemsNumber Theory60F05, 11F27, 60B10, 11J70, 37E05, 28D05text