The first zero of an empirical characteristic function (Q923472)

The stochastic process \((U_ n(t)\), \(t\geq 0)\) defined by \(U_ n(t)=n^{-1}\sum^{n}_{j=1}\cos (tX_ j)\) is the real part of the empirical characteristic function, generated by the sample \(X_ 1,...,X_ n\). The authors derive the limiting distribution of \(R_ n=\inf \{t>0\); \(U_ n(t)=0\}\). Different cases are considered depending on \(u(t)=E \cos (tX)\), and it is shown that possible limit distributions are normal and extreme value distributions.