EXPECTED NUMBER OF REAL ZEROS OF A CLASS OF RANDOM HYPERBOLIC POLYNOMIAL

Mean Number of Real Zeros of a Random Hyperbolic Polynomial

Consider the random hyperbolic polynomial, f(x) = 1a1 coshx+···+np × an coshnx, in which n and p are integers such that n ≥ 2, p ≥ 0, and the coefficients ak(k = 1,2, . . . ,n) are independent, standard normally distributed random variables. If νnp is the mean number of real zeros of f(x), then we prove that νnp = π−1 logn+ O{(logn)1/2}.

On the Expected Number of Zeros of a Random Harmonic Polynomial

We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

Expected Number of Local Maxima of Some Gaussian Random Polnomials

On Expected Number of Level Crossings of a Random Hyperbolic Polynomial

Let g1(ω), g2(ω), . . . , gn(ω) be independent and normally distributed random variables with mean zero and variance one. We show that, for large values of n, the expected number of times the random hyperbolic polynomial y = g1(ω) coshx + g2(ω) cosh 2x + · · · + gn(ω) coshnx crosses the line y = L, where L is a real number, is 1 π logn+ O(1) if L = o( √ n) or L/ √ n = O(1), but decreases steadi...

Covariance of the number of real zeros of a random trigonometric polynomial

For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ) = ∑n j=0{aj cos jθ + bj sin jθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/ √ 3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value f...