Stationary Gaussian processes with a finite correlation function

If the correlation function vanishes outside the segment [-R, R], then an upper estimate (uniform with respect to all such processes) is possible for the probability of the fact that on an other segment [-r, r] the process remains between - ε and ε. Such an estimate is obtained, decreasing for ε → 0 as exp(-f(r/Rln²+ ∞) and, moreover, r/R may be either 0 or +∞. The proof is based on an estimate of the form ∥ P_mQ_n ∥ ≥ c_mn ∥ P_m∥{dot operator} ∥ Q_n ∥ for norms of polynomials on a circle in the complex plane.