Abstract
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 asexp(−f(r/R ln 2+ ∞) and, moreover,r/R may be either 0 or +∞. The proof is based on an estimate of the form ∥ PmQn ∥ ≥ cmn ∥ Pm∥⋅ ∥ Qn ∥ for norms of polynomials on a circle in the complex plane.
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References
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 279–288, 1990.
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Tsirel'son, B.S. Stationary Gaussian processes with a finite correlation function. J Math Sci 68, 597–603 (1994). https://doi.org/10.1007/BF01254288
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DOI: https://doi.org/10.1007/BF01254288