The real zeros of a random algebraic polynomial with dependent coefficients

The -level crossings of a random algebraic polynomial with dependent coefficients

“…where now the expression on the right is the expected number of real zeros of P n (x) on (−∞, −1). Thus, Theorem 1.1 in [6] yields the upper bounds…”

“…which are derived in the proof of Lemma 3.1 in [6]. Combining (3.6) with (3.1), (3.2), and (3.3), and after some tedious algebra, we can obtain the expression…”

“…Notice that this upper bound holds on the entire interval (−1, 1). Next, from equations (3.5) and (3.7) in [6] we know that…”

“…where f (φ) is the spectral density of the covariance function (in addition to the discussion in [6], see [1] and [2] for further references). By imposing certain conditions on the spectral density, for the random polynomial P n (x) given by (1.1), we will be able to study the level crossings for a wide range of covariance functions.…”

“…These results will be proved using the techniques developed by Farahmand in [3] and [4], as well as the spectral density of the covariance function. We will also make use of several results from [6], which in turn draws heavily from the work of Sambandham in [7]. Letting N K (α, β) be the number of K-level crossings of P n (x) in the interval (α, β), the main theorem is formulated as follows.…”