The maximum of the periodogram

Let x(t), t = 1,…, T, be generated by a zero mean stationary process and let $\text{I(ω) =}\text{|Σx(t)}\text{exp}\text{itω|}{}^2\text{T}$ be the periodogram. Under general conditions, and in particular assuming only a finite 2nd moment, it is shown that $\text{max}{}_{\mathrm{\omega }}\text{I(ω)}\text{{2πf(ω)}\text{log}\text{T}}\text{≤ 1}$, a.s., and under stricter conditions it is shown that equality holds.