Uniform convergence of empirical characteristic functions in a complex domain with applications to option pricing

In Csörgő and Totik (1983) and Csörgő (1985) it has been shown that in the case of independent identically distributed (iid) random variables X1,X2,…,XnX1,X2,…,Xn the empirical characteristic function (ecf) ϕˆn(u) converges uniformly, for |u|≤Un|u|≤Un to the characteristic function ϕ(u)ϕ(u) of XX, on increasing intervals which union covers the whole real line. We show that if suitable moments exist then the uniform convergence is also valid for uu in the complex domain x=u+iν,|u|≤Un,ν∈(a,b), where a