Asymptotic properties of the spectral test, diaphony, and related quantities (Q2759098)

In this very interesting paper the author presents the limit laws of discrepancies defined via exponential sums, and algorithms (with error bounds) to approximate the corresponding cumulative distribution functions.NEWLINENEWLINENEWLINEIn Section 2, the limits of various discrepancies are derived. The results apply to the weighted and the nonweighted spectral test of \textit{P. Hellekalek} [Lect. Notes Stat. 138, 49-108 (1998; Zbl 0937.65004)] and various instances of general discrepancies of \textit{F. J. Hickernell} [Math. Comput. 67, No. 221, 299-322 (1998; Zbl 0889.41025); Lect. Notes Stat., Springer-Verlag 138, 109-166 (1998; Zbl 0920.65010)] and \textit{J. K. Hoogland} and \textit{R. Kleiss} [Comput. Phys. Commun. 98, No. 1-2, 111-127 (1996; Zbl 0926.65027)] for the exponential function system, as well as classical quantities like \textit{R. R. Coveyou} and \textit{R. D. MacPherson's} spectral test [J. Assoc. Comput. Mach. 14, 100-119 (1967; Zbl 0155.22801)], \textit{P. Zinterhof's} diaphony [Österr. Akad. Wiss., Math.-naturw. Kl., S.-Ber., Abt. II 185, 121-132 (1976; Zbl 0356.65007)] and \textit{P. Zinterhof} and \textit{H. Stegbuchner} [Stud. Sci. Math. Hung. 13, 273-289 (1978; Zbl 0458.42003)], and the Zaremba figure of merit.NEWLINENEWLINENEWLINEIn Section 3, the approximations and approximation error bounds are obtained.NEWLINENEWLINENEWLINEThe work contains a bibliography including 25 references from the most significant works in this domain.