On the Expected Number of Level Crossings of Random Trigonometric Polynomials

Abstract

This paper provides an asymptotic estimate for the expected number of level-crossings of random trigonometric polynomials a ₀(ω) + a ₁(ω)cosΘ + ··· + a _n (ω)cos n Θ, where a _j (ω), j = 0,…, n are k groups of independent normally distributed random variables with means μ _i and variances , i = 1,…, k. It is shown that in order to obtain our result we need charac-teristics, namely μ _i and , i = 1,…, k, to be any bounded absolute constants.

Keywords:

• Jensen's theorem
• Kac–Rice formula
• Level crossings
• Number of real zeros
• Random polynomials

Mathematics Subject Classification:

• Primary 60G99
• Secondary 60H99