International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 189675 | https://doi.org/10.1155/2008/189675

K. Farahmand, A. Grigorash, B. McGuinness, "On Different Classes of Algebraic Polynomials with Random Coefficients", International Journal of Stochastic Analysis, vol. 2008, Article ID 189675, 8 pages, 2008. https://doi.org/10.1155/2008/189675

On Different Classes of Algebraic Polynomials with Random Coefficients

Academic Editor: Lev Abolnikov
Received27 Feb 2008
Accepted20 Apr 2008
Published22 May 2008

Abstract

The expected number of real zeros of the polynomial of the form ğ‘Ž0+ğ‘Ž1𝑥+ğ‘Ž2𝑥2+⋯+ğ‘Žğ‘›ğ‘¥ğ‘›, where ğ‘Ž0,ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘› is a sequence of standard Gaussian random variables, is known. For 𝑛 large it is shown that this expected number in (−∞,∞) is asymptotic to (2/𝜋)log𝑛. In this paper, we show that this asymptotic value increases significantly to √𝑛+1 when we consider a polynomial in the form ğ‘Ž0𝑛01/2√𝑥/1+ğ‘Ž1𝑛11/2𝑥2/√2+ğ‘Ž2𝑛21/2𝑥3/√3+⋯+ğ‘Žğ‘›î‚€ğ‘›ğ‘›î‚1/2𝑥𝑛+1/√𝑛+1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.

1. Introduction

The classical random algebraic polynomial has previously been defined as𝑇(𝑥)≡𝑇𝑛(𝑥,𝜔)=𝑛𝑗=0ğ‘Žğ‘—(𝜔)𝑥𝑗,(1.1)where, for (Ω,𝒜,Pr) a fixed probability space, {ğ‘Žğ‘—(𝜔)}𝑛𝑗=0 is a sequence of independent random variables defined on Ω. For 𝑛 large, the expected number of real zeros of 𝑇(𝑥), in the interval (−∞,∞), defined by EN0,𝑇(−∞,∞), is known to be asymptotic to (2/𝜋)log𝑛. For this case the coefficients ğ‘Žğ‘—â‰¡ğ‘Žğ‘—(𝜔) are assumed to be identical normal standard. This asymptotic value was first obtained by the pioneer work of Kac [1] and was recently significantly improved by Wilkins [2], who reduced the error term involved in this asymptotic formula to 𝑂(1). Since then, many other mathematical properties of 𝑇(𝑥) have been studied and they are listed in [3] and more recently in [4].

The other class of random polynomials is introduced in an interesting article of Edelman and Kostlan [5] in which the 𝑗th coefficients of 𝑇(𝑥) in (1.1) have nonidentical variance (𝑛𝑗). It is interesting to note that in this case the expected number of zeros significantly increased to √𝑛, showing that the curve representing this type of polynomial oscillates significantly more than the classical polynomial (1.1) with identical coefficients. As it is the characteristic of (𝑛𝑗), 𝑗=0,1,2,…,𝑛 maximized at the middle term of 𝑗=[𝑛/2], it is natural to conjecture that for other classes of distributions with this property the polynomial will also oscillate significantly more. This conjecture is examined in [6, 7]. This interesting and unexpected property of the latter polynomial has its close relation to physics reported by Ramponi [8], which together with its mathematical interest motivated us to study the polynomial𝑃(𝑥)≡𝑃𝑛(𝑥,𝜔)=𝑛𝑗=0ğ‘Žğ‘—îƒ©ğ‘›ğ‘—îƒª1/2𝑥𝑗+1√𝑗+1.(1.2)As we will see, because of the presence of the binomial elements in (1.2), we can progress further than the classical random polynomial defined in (1.1). However, even in this case the calculation yields an asymptotic result rather than equality. With a small change to the definition of the polynomial we show that the result improves. To this end we define𝑄(𝑥)≡𝑄𝑛(𝑥,𝜔)=𝑛𝑗=0ğ‘Žğ‘—îƒ©ğ‘›ğ‘—îƒª1/2𝑥𝑗+1√+ğ‘Žğ‘—+1∗√𝑛+1,(1.3)where ğ‘Žâˆ— is mutually independent of and has the same distribution as {ğ‘Žğ‘—}𝑛𝑗=0. We prove the following.

Theorem 1.1. When the coefficients ğ‘Žğ‘— of 𝑃(𝑥) are independent standard normal random variables, then the expected number of real roots is asymptotic to 𝐸𝑁0,𝑃√(−∞,∞)∼𝑛+1.(1.4)

Corollary 1.2. With the same assumption as Theorem 1.1 for the coefficients ğ‘Žğ‘— and ğ‘Žâˆ— one has 𝐸𝑁0,𝑄√(−∞,∞)=𝑛+1.(1.5)

Also of interest is the expected number of times that a curve representing the polynomial cuts a level 𝐾. We assume 𝐾 is any constant such that(i)𝐾2≤𝑒𝑛𝑛2,1(ii)𝑛2𝐾=𝑜2,(iii)𝐾2𝑒=𝑜𝑛𝑛2.(1.6)For example, any absolute constant 𝐾≠0 satisfies these conditions. Defining EN𝐾,𝑃 as the expected number of real roots of 𝑃(𝑥)=𝐾, we can generalize the above theorem to the following one.

Theorem 1.3. When the coefficients ğ‘Žğ‘— have the same distribution as in Theorem 1.1, and 𝐾 obeys the above conditions (i)–(iii), the asymptotic estimate for the expected number of K-level crossings is 𝐸𝑁𝐾,𝑄(−∞,∞)∼𝐸𝑁𝐾,𝑃√(−∞,∞)∼𝑛+1.(1.7)

The other characteristic which also gives a good indication of the oscillatory behavior of a random polynomial is the expected number of maxima or minima. We denote this expected number by EN𝑀𝑃 for polynomial 𝑃(𝑥) given in (1.2) and, since the event of tangency at the 𝑥-axis has probability zero, we note that this is asymptotically the same as the expected number of real zeros of ğ‘ƒî…ž(𝑥)=𝑑𝑃(𝑥)/𝑑𝑥. In the following theorem, we give the expected number of maxima of the polynomial.

Theorem 1.4. With the above assumptions on the coefficients ğ‘Žğ‘—, then the asymptotic estimate for the expected number of maxima of 𝑃(𝑥) is 𝐸𝑁𝑀𝑃√(−∞,∞)∼𝑛.(1.8)

Corollary 1.5. With the above assumptions for the coefficients ğ‘Žğ‘— and ğ‘Žâˆ— one has 𝐸𝑁𝑀𝑄√(−∞,∞)∼𝑛.(1.9)

2. Proof of Theorem 1.1

We use a well-known Kac-Rice formula, [1, 9], in which it is proved thatEN0,𝑃(ğ‘Ž,𝑏)=ğ‘ğ‘ŽÎ”ğœ‹ğ´2𝑑𝑥,(2.1)where ğ‘ƒî…ž(𝑥) represents the derivative with respect to 𝑥 of 𝑃(𝑥). We denote𝐴2=var𝑃(𝑥),𝐵2𝑃=var(𝑥),𝐶=cov𝑃(𝑥),ğ‘ƒî…žî‚(𝑥),Δ2=𝐴2𝐵2−𝐶2.(2.2)Now, with our assumptions on the distribution of the coefficients, it is easy to see that𝐴2=𝑛𝑗=0𝑛𝑗𝑥2𝑗+2=𝑗+11+𝑥2𝑛+1−1𝑛+1,𝐵𝑛+1(2.3)2=𝑛𝑗=0𝑛𝑗(𝑗+1)𝑥2𝑗=1+𝑥2𝑛−11+𝑥2+𝑛𝑥2,(2.4)𝐶=𝑛𝑗=0𝑛𝑗𝑥2𝑗+1=𝑥1+𝑥2𝑛.(2.5) We note that, for all sufficiently large 𝑛 and 𝑥 bounded away from zero, from (2.3) we have𝐴2∼1+𝑥2𝑛+1𝑛+1.(2.6)This together with (2.1), (2.4), and (2.5) yieldsEN02(−∞,∞)∼𝜋𝜖0Δ𝐴22𝑑𝑥+ğœ‹î€œâˆžğœ–âˆšğ‘›+11+𝑥2𝑑𝑥,(2.7)where 𝜖>0, 𝜖→0 as ğ‘›â†’âˆž. The second integral can be expressed as2√𝑛+1𝜋𝜋2→√−arctan𝜖𝑛+1asğ‘›â†’âˆž.(2.8)In the first integral, the expression (Δ/𝐴2) has a singularity at 𝑥=0:Δ𝐴2=⎷(𝑛+1)1+𝑥22𝑛−1+𝑥2𝑛−11+𝑥2+𝑛𝑥21+𝑥2𝑛+1−12.(2.9)Notice that the expression in (2.9) is bounded from above:Δ𝐴2<√(𝑛+1)(1−𝐷)1+𝑥2,(2.10)where𝐷=1+𝑛𝑥21+𝑥2𝑛−1−1+𝑥2𝑛1+𝑥2𝑛−12=(𝑛−1)1+𝑥2𝑛−2+(𝑛−2)1+𝑥2𝑛−3+⋯+31+𝑥22+21+𝑥2+11+𝑥2𝑛−1+1+𝑥2𝑛−2+⋯+1+𝑥2+12.(2.11) When 𝑥=0, we have𝑛𝐷=2−𝑛2𝑛2(2.12)and therefore Δ𝐴2<𝑛+1√∼2𝑛𝑛+12,(2.13)which means that the integrand in the first integral of (2.7) is bounded for every 𝑛. When 𝑥>0, it can easily be seen that∑1>𝐷>𝑛−2𝑗=0(1+𝑗)𝑛21+𝑥22𝑛−2>0,(2.14)and thereforeΔ𝐴2<√𝑛+11+𝑥2.(2.15)Hence, the first integral that appears in (2.7) is bounded from above as follows:2𝜋𝜖0Δ𝐴22𝑑𝑥<𝜋𝜖0√𝑛+11+𝑥2√𝑑𝑥=2(arctan𝜖)𝑛+1𝜋√=𝑜𝑛+1(2.16)by the choice of 𝜖. Altogether, the value of the first integral in (2.7) is of a smaller order of magnitude than the value of the second integral, and we have from (2.7)EN0√(−∞,∞)∼𝑛+1(2.17)which completes the proof of Theorem 1.1.

In order to obtain the proof of Corollary 1.2, we note that the above calculations remain valid for 𝐵2 and 𝐶. However, for 𝐴2 we can obtain the exact value rather than the asymptotic value. To this end, we can easily see that𝐴2𝑄==var𝑄(𝑥)𝑛𝑗=0𝑛𝑗𝑥2𝑗+2+1𝑗+1=𝑛+11+𝑥2𝑛+1𝑛+1.(2.18)Substituting this value instead of (2.3) together with (2.4) and (2.5) in the Kac-Rice formula (2.1), we get a much more straight forward expression than that in the above proof:EN0,𝑄1(−∞,∞)=ğœ‹î€œâˆž0√𝑛+11+𝑥2√𝑑𝑥=𝑛+1.(2.19)This gives the proof of Corollary 1.2.

3. Level Crossings

To find the expected number of 𝐾-level crossings, we use the following extension to the Kac-Rice formula as it was used in [10]. It is shown that in the case of normal standard distribution of the coefficientsEN𝐾(ğ‘Ž,𝑏)=𝐼1(ğ‘Ž,𝑏)+𝐼2(ğ‘Ž,𝑏)(3.1)with 𝐼1(ğ‘Ž,𝑏)=ğ‘ğ‘ŽÎ”ğœ‹ğ´2−𝐵exp2𝐾22Δ2𝐼𝑑𝑥,(3.2)2(ğ‘Ž,𝑏)=ğ‘ğ‘Žâˆš2𝐾𝐶𝜋𝐴3−𝐾exp22𝐴2−erf𝐾𝐶√2𝐴Δ𝑑𝑥,(3.3) where, as usual, ∫erf(𝑥)=𝑥0√exp(−𝑡)𝑑𝑡≤𝜋/2. Since changing 𝑥 to −𝑥 leaves the distribution of the coefficients unchanged, EN𝐾(−∞,0)=EN𝐾(0,∞). Hence to what follows we are only concerned with 𝑥≥0. Using (2.3)–(2.5) and (3.2) we obtain𝐼12√(−∞,∞)=𝑛+1ğœ‹î€œâˆž011+𝑥2−𝐾exp2(𝑛+1)1+𝑥2+𝑛𝑥221+𝑥2𝑛+1𝑑𝑥.(3.4)Using substitution 𝑥=tan𝜃 in (3.4) we can see that𝐼1(−∞,∞)=𝐽1𝜋0,2=2√𝑛+1𝜋0𝜋/2exp−𝐾2(𝑛+1)21+𝑛sin2𝜃cos2𝑛𝜃𝑑𝜃,(3.5)where the notation 𝐽1 emphasizes integration in 𝜃. In order to progress with the calculation of the integral appearing in (3.5), we first assume 𝜃>𝛿, where 𝛿=arccos(1−1/(𝑛𝜖)), where 𝜖=1/{2log(𝑛𝐾)}. This choice of 𝜖 is indeed possible by condition (i). Now since cos𝜃<(1−1/(𝑛𝜖)), we can show thatcos2𝑛1𝜃<1−𝑛𝜖2𝑛=11−𝑛𝜖−𝑛𝜖−2/𝜖−2∼exp𝜖→0(3.6)as ğ‘›â†’âˆž. Now we are in a position to evaluate the dominated term which appears in the exponential term in (3.5). From (3.6), it is easy to see that for our choice of 𝜃𝐾2𝑛2cos2𝑛−2𝜃<𝐾2𝑛2−2exp𝜖=𝐾2𝑛2exp−4log(𝑛𝐾)=(𝐾𝑛)−2→0,(3.7)by condition (ii). Therefore, for all sufficiently large 𝑛, the argument of the exponential function in (3.5) is reduced to zero, and hence the integrand is not a function of 𝜃 and we can easily see by the bounded convergence theorem and condition (iii) that𝐽1𝜋𝛿,2∼√𝑛+1.(3.8)Since the argument of the exponential function appearing in (3.5) is always negative, it is straight forward for our choice of 𝛿 and 𝜖 to see that𝐽12√(0,𝛿)<𝑛+1𝜋𝛿02𝑑𝜃=𝜋√𝑛+1arccos1−2log(𝑛𝐾)𝑛√=𝑜(𝑛+1),(3.9)by condition (iii). As 𝐼1(−∞,∞)=𝐽1(0,𝛿)+𝐽1(𝛿,𝜋/2), by (3.8) and (3.9) we see that𝐼1√(−∞,∞)∼𝑛+1.(3.10)Now we obtain an upper limit for 𝐼2 defined in (3.3). To this end, we let √𝑣=𝐾/(2𝐴). Then we have𝐼2||𝐾||(−∞,∞)≤√2ğœ‹âˆžâˆ’âˆžğ¶ğ´3−𝐾exp22𝐴22𝑑𝑥=âˆšğœ‹î€œâˆž0exp−𝑣22𝑑𝑣≤√𝜋.(3.11)This together with (3.10) proves that EN𝐾,𝑄√(−∞,∞)∼𝑛+1. The theorem is proved for polynomial 𝑄(𝑥) given in (1.3).

Let us now prove the theorem for polynomial 𝑃(𝑥) given in (1.2), that isEN𝐾,𝑃√(−∞,∞)∼𝑛+1.(3.12)The proof in this case repeats the proof for EN𝐾,𝑄(−∞,∞) above, except that the equivalent of (3.4) will be an asymptotic rather than an exact equality, and the derivation of the equivalent of (3.9) is a little more involved, as shown below. Going back from the new variable 𝜃 to the original variable 𝑥 gives𝐽12(0,𝛿)=𝜋0tan𝛿Δ𝐴2−𝐵exp2𝐾22Δ22𝑑𝑥<𝜋0tan𝛿Δ𝐴2𝑑𝑥,(3.13)where Δ/𝐴2 is given by (2.9). Then by the same reasoning as in the proof of Theorem 1.1,𝐽12√(0,𝛿)<𝑛+1𝜋2√arctan(tan𝛿)=𝑛+1𝜋𝛿=2√𝑛+1𝜋arccos1−2log(𝑛𝐾)𝑛√=𝑜,𝑛+1(3.14)by condition (iii). This completes the proof of Theorem 1.3.

4. Number of Maxima

In finding the expected number of maxima of 𝑃(𝑥), we can find the expected number of zeros of its derivative ğ‘ƒî…ž(𝑥). To this end we first obtain the following characteristics needed in order to apply them into the Kac-Rice formula (2.1),𝐴2𝑀𝑃=var=(𝑥)𝑛𝑗=0𝑛𝑗(𝑗+1)𝑥2𝑗=1+𝑥2𝑛−11+𝑥2+𝑛𝑥2,𝐵(4.1)2𝑀=var(ğ‘ƒî…žî…žî‚=(𝑥)𝑛𝑗=0𝑛𝑗𝑗2(𝑗+1)𝑥2𝑗−2=𝑛1+𝑥2𝑛−32+4𝑛𝑥2+𝑛𝑥4+𝑛2𝑥4,𝐶(4.2)𝑀𝑃=cov(𝑥),ğ‘ƒî…žî…žî‚=(𝑥)𝑛𝑗=0𝑛𝑗𝑗(𝑗+1)𝑥2𝑗−1=𝑛𝑥1+𝑥2𝑛−22+𝑥2+𝑛𝑥2.(4.3) Hence from (4.1)–(4.3) we obtainΔ2𝑀=𝐴2𝑀𝐵2𝑀−𝐶2𝑀=𝑛1+𝑥22𝑛−42+𝑛𝑥4+𝑛2𝑥4+2𝑥2+2𝑛𝑥2.(4.4)Now from (4.1) and (4.5) we haveΔ𝑀𝐴2𝑀=𝑛2+𝑛𝑥4+𝑛2𝑥4+2𝑥2+4𝑛𝑥21+𝑥21+𝑥2+𝑛𝑥2.(4.5)As the value of 𝑥 increases, the dominating terms in (4.5) change. For accuracy therefore, the interval needs to be broken up. In this case, the interval (0,∞) was divided into two subintervals. First, choose 𝜖<𝑥<∞ such that 𝜖=𝑛−1/4, thenΔ𝑀𝐴2𝑀∼√𝑛1+𝑥2.(4.6)Substituting into the Kac-Rice formula (2.1) yieldsEN𝑀𝑃1(𝜖,∞)âˆ¼ğœ‹î€œâˆžğœ–âˆšğ‘›1+𝑥2√𝑑𝑥=𝑛2.(4.7)Now we choose 0<𝑥<𝜖. Since for 𝑛 sufficiently large the term 𝑛2𝑥4 is significantly larger than 𝑛𝑥4 and also since for this range of 𝑥 we can see 2𝑥2<1, we can obtain an upper limit for (4.5) asΔ𝑀𝐴2𝑀<𝑛3+2𝑛2𝑥4+4𝑛𝑥21+𝑛𝑥2<𝑛3+6𝑛𝑥2+3𝑛2𝑥41+𝑛𝑥2=√3𝑛.(4.8)Substituting this upper limit into Kac-Rice formula, we can seeEN𝑀𝑃(0,𝜖)=𝜖0Δ𝑀𝜋𝐴2𝑀√𝑑𝑥<𝑛3𝑛𝜖=𝑜1/4.(4.9)This together with (4.7) completes the proof of Theorem 1.4. To prove Corollary 1.5, it suffices to notice that since ğ‘„î…ž(𝑥)=ğ‘ƒî…ž(𝑥) and ğ‘„î…žî…ž(𝑥)=ğ‘ƒî…žî…ž(𝑥), all the arguments in the above proof apply to polynomial 𝑄(𝑥), and we have therefore EN𝑀𝑃(ğ‘Ž,𝑏)=EN𝑀𝑄(ğ‘Ž,𝑏).

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Copyright © 2008 K. Farahmand 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.


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