Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2015, Article ID 261370, 7 pages http://dx.doi.org/10.1155/2015/261370 Research Article Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients Mina Ketan Mahanti,1 Amandeep Singh,2 and Lokanath Sahoo3 1 Department of Mathematics, College of Basic Science and Humanities, OUAT, Bhubaneswar, India DPS Kalinga, Bhubaneswar, India 3 Gopabandhu Science College, Athagad, India 2 Correspondence should be addressed to Mina Ketan Mahanti; minaketan Received 7 January 2015; Revised 21 May 2015; Accepted 27 May 2015 Academic Editor: Niansheng Tang Copyright © 2015 Mina Ketan Mahanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form 𝑦 = 𝑃𝑛 (𝑡) = √( 𝑛1 )𝑎1 cosh 𝑡 + √( 𝑛2 )𝑎2 cosh 2𝑡 + ⋅ ⋅ ⋅ + √( 𝑛𝑛 )𝑎𝑛 cosh 𝑛𝑡, where 𝑎1 , . . . , 𝑎𝑛 is a sequence of standard Gaussian random variables, is √𝑛/2 + 𝑜𝑝 (1). It is shown that the asymptotic value of expected number of times the polynomial crosses the level y = K is also √𝑛/2 as long as 𝐾 does not exceed √2𝑛 𝑒𝜇(𝑛) , where 𝜇(𝑛) = 𝑜(𝑛). The number of oscillations of 𝑃𝑛 (𝑡) about 𝑦 = 𝐾 will be less than √𝑛/2 asymptotically only if 𝐾 = √2𝑛 𝑒𝜇(𝑛) , where 𝜇(𝑛) = 𝑂(𝑛) or 𝑛−1 𝜇(𝑛) → ∞. In the former case the number of oscillations continues to be a fraction of √𝑛 and decreases with the increase in value of 𝜇(𝑛). In the latter case, the number of oscillations reduces to 𝑜𝑝 (√𝑛) and almost no trace of the curve is expected to be present above the level 𝑦 = 𝐾 if 𝜇(𝑛)/(𝑛 log 𝑛) → ∞. 1. Introduction 𝑗=𝑛 Let (Ω, 𝐴, Pr) be a fixed probability space and let {𝑎𝑘 (𝜔)}𝑘=1 be a sequence of independent random variables defined on Ω. The sum 𝑎0 (𝜔)𝑓1 (𝑡) + 𝑎2 (𝜔)𝑓2 (𝑡) + ⋅ ⋅ ⋅ + 𝑎𝑛 (𝜔)𝑓𝑛 (𝑡) is traditionally known as a random algebraic polynomial if 𝑓𝑖 (𝑡) = 𝑡𝑖 , a random trigonometric polynomial if 𝑓𝑖 (𝑡) = cos(𝑖𝑡) or sin(𝑖𝑡), and a random hyperbolic polynomial if 𝑓𝑖 (𝑡) = cosh(𝑖𝑡) or sinh(𝑖𝑡). One can have useful information about the behaviour of these ensembles of polynomials if the average number of times these polynomials oscillate about the line 𝑦 = 𝐾 is known. The reader is referred to the book by Farahmand [1] where an exhaustive account of progress made in study of random polynomials has been presented. It is to be noted that there is significantly more published literature on random algebraic and random trigonometric polynomials than that of random hyperbolic polynomials. Let 𝑛 𝑦 = 𝑄𝑛 (𝑡) = ∑ 𝑎𝑘 (𝜔) cosh 𝑘𝑡, 𝑘=1 (1) where 𝑎𝑘 (𝜔) are normally distributed random variables with mean zero and variance one. One knows that Das [2] first calculated the expected number of real zeros of 𝑄𝑛 (𝑡). Farahmand [3] calculated the asymptotic estimate of oscillations of 𝑄𝑛 (𝑡) about 𝑦 = 𝐾 if 𝐾 = 𝑜(√𝑛). Some of the other works in this direction are due to Mahanti [4–6]. Wilkins [7] determined real zeros of 𝑄𝑛 (𝑡) when var(𝑎𝑘 (𝜔)) = 𝑘𝑝 , 𝑝 ≥ 0. We observe that the asymptotic value of the oscillations of random hyperbolic polynomials is (2/𝜋) log 𝑛 in each of these cases. One is tempted to ask whether 𝑄𝑛 (𝑡) has more than (2/𝜋) log 𝑛 oscillations under certain conditions. In this context, we are reminded of a recent work of Edelman and Kostlan [8] where it has been found out that the expected number of real zeros of random algebraic polynomials increases significantly if the variance of the coefficients changes from unity to √( 𝑛𝑘 ). Therefore, we examine what effect this new assumption on variance of the coefficients has on number of 2 International Journal of Mathematics and Mathematical Sciences −1 (v) EN𝑛,𝐾 (−∞, ∞) = √𝑛𝑒−𝑛 𝜇(𝑛) /(2√2) + 2/√𝜋 + 𝑂𝑝 (𝑛−1/2 log 𝑛) if 𝜇(𝑛)/𝑛 → ∞ and 𝐾 = √2𝑛 𝑒𝜇(𝑛) . oscillations of 𝑄𝑛 (𝑡). In other words, we calculate the number of oscillations of the polynomial 𝑛 𝑛 𝑦 = 𝑃𝑛 (𝑡) = ∑ 𝑎𝑘 (𝜔) √ ( ) cosh 𝑘𝑡, 𝑘 𝑘=1 (2) where 𝑎𝑘 (𝜔) are normally distributed random variables with mean zero and variance one. In Theorem 1 we have shown that the number of real zeros of 𝑃𝑛 (𝑡) is substantially larger than that of 𝑄𝑛 (𝑡). Moreover, there is a significant difference in the way real zeros of 𝑃𝑛 (𝑡) and 𝑄𝑛 (𝑡) lie on the 𝑡-axis. Most of the real zeros of 𝑄𝑛 (𝑡) are confined to the interval [−1, 1] and there are negligible numbers of them if |𝑡| > 1 (see [4]). But there are large numbers of real zeros of 𝑃𝑛 (𝑡) outside [−1, 1]. This phenomenon can be deduced from the formula given in (26) and Lemma 10. In fact, the number of real zeros of 𝑃𝑛 (𝑡) in the region |𝑡| > 𝛼, 𝛼 > 1/√𝑛, is dependent on 𝛼. The real zeros decrease in number with increase in 𝛼 in the region |𝑡| > 𝛼 and there are negligible numbers of them only when 𝛼 → ∞. Let 𝜇(𝑛) be any function of 𝑛 such that 𝜇(𝑛)/𝑛 → 0. In Theorem 2 we have shown that the number of oscillations of 𝑃𝑛 (𝑡) about the line 𝑦 = 𝐾, where 𝐾 ≤ √2𝑛 𝑒𝜇(𝑛) , is equal to its axis crossings. Thus, for all these values of 𝐾 one can say that most of the oscillations of 𝑃𝑛 (𝑡) that cross the 𝑡-axis reach up to the level 𝑦 = 𝐾. If 𝜇(𝑛) = 𝑂(𝑛) or 𝑛−1 𝜇(𝑛) → ∞, we have proved in the theorem that the asymptotic value of number of oscillations of 𝑃𝑛 (𝑡) about 𝑦 = 𝐾 = √2𝑛 𝑒𝜇(𝑛) is less than √𝑛/2. If 𝜇(𝑛) = 𝑂(𝑛) the number of oscillations continues to be a fraction of √𝑛 and decreases with the increase in value of 𝜇(𝑛). If 𝑛−1 𝜇(𝑛) → ∞ the number of oscillations is reduced to 𝑜𝑝 (√𝑛). Inequality (35) together with Lemma 14 provides a glimpse of the manner in which the number of oscillations decreases with increase in value of 𝐾. Inequality (35) also shows that there is hardly any trace of the curve 𝑃𝑛 (𝑡) above the level 𝑦 = 𝐾 = √2𝑛 𝑒𝜇(𝑛) if 𝜇(𝑛)/(𝑛 log 𝑛) → ∞. Let the coefficients {𝑎𝑘 (𝜔)}𝑘=𝑛 𝑘=1 of 𝑃𝑛 (𝑡) be standard normal random variables. The expected number of oscillations of 𝑃𝑛 (𝑡) about the line 𝑦 = 𝐾, 𝑡 ∈ (𝛼, 𝛽), has been denoted by us as EN𝑛,𝐾 (𝛼, 𝛽) in the following two theorems. Two more differences in behaviour of 𝑃𝑛 (𝑡) and 𝑄𝑛 (𝑡) are noteworthy. In what follows we will find out that most of the axis crossings of 𝑃𝑛 (𝑡) reach the level √2𝑛 𝑒𝜇(𝑛) . However, the branches of 𝑄𝑛 (𝑡) that cross the axis do not travel beyond 𝑦 = 𝐾, where 𝐾 = 𝑂(√𝑛) (see Mahanti and Sahoo [6]). Almost all of the polynomial 𝑄𝑛 (𝑡) lie below the level 𝐾, where 𝑛−1 log(𝐾/√𝑛) → ∞ (Mahanti and Sahoo [6]). However, a large part of 𝑃𝑛 (𝑡) stretches above this level. 2. Formula for the Proof of the Theorems The proof of the theorem is based on the formula for expected number of level crossings given by Crammer and Leadbetter [9, page 285]. Using it for 𝑃𝑛 (𝑡) − 𝐾 = 0 in the interval (𝛼, 𝛽) we can show that EN𝑛,𝐾 (𝛼, 𝛽) = ∫ 𝛼 √𝑛 + 𝑂𝑝 (𝑛−1/2 log 𝑛) . 2 Theorem 2. For sufficiently large 𝑛, −1/2 −1/2 (iii) EN𝑛,𝐾 (−∞, ∞) = √𝑛/2 − 𝑛 𝜇(𝑛)/(2𝜋) + 2/√𝜋 + 𝑂 (...truncated)