Global zero-count asymptotics for random superpositions of triangular signals

Determine whether the random superposition of triangular signals has the same normalized expected number of zeros per period as the universal limit established for piecewise-polynomial, globally -regular base functions, namely whether its expected zero count divided by the degree converges to the corresponding universal constant.

Background

The paper proves global universality for the expected number of zeros of random periodic signals of the form $f_n(t)=n^{-1/2}\sum_{k=1}^n a_k f(kt)$ when the base function $f$ satisfies conditions (H1) and (H2). In particular, (H2) requires $f$ to be piecewise polynomial and globally of class $\mathcal C^7$, together with non-vanishing conditions on its derivatives.

The triangular wave is a motivating example of a non-analytic base function, but its available regularity does not permit the authors to apply their equi-integrability argument for the normalized number of zeros. Since this equi-integrability is needed to upgrade convergence in distribution to convergence of first moments, the universal asymptotic zero-count limit for random superpositions of triangular signals remains unresolved.