International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 725260 | https://doi.org/10.1155/2009/725260

K. Farahmand, M. Sambandham, "Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions", International Journal of Stochastic Analysis, vol. 2009, Article ID 725260, 6 pages, 2009. https://doi.org/10.1155/2009/725260

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

Academic Editor: Lev Abolnikov
Received28 Jan 2009
Accepted26 Feb 2009
Published07 Apr 2009

Abstract

The expected number of real zeros of an algebraic polynomial ğ‘Žğ‘œ+ğ‘Ž1𝑥+ğ‘Ž2𝑥2+⋯+ğ‘Žğ‘›ğ‘¥ğ‘› with random coefficient ğ‘Žğ‘—,𝑗=0,1,2,…,𝑛 is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the 𝑗th coefficient is var(ğ‘Žğ‘—î€·)=𝑛𝑗. It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume 𝐸(ğ‘Žğ‘—î€·)=𝑛𝑗𝜇𝑗+1 and var(ğ‘Žğ‘—î€·)=ğ‘›ğ‘—î€¸ğœŽ2𝑗. We show how the above expected number of real zeros is dependent on values of ğœŽ2 and 𝜇 in various cases.

1. Origin of Polynomials

Let (Ω,Pr,𝒜) be a fixed probability space, and for 𝜔∈Ω let {ğ‘Žğ‘—(𝜔)}𝑛𝑗=0 be a sequence of independent identically distributed random variables defined on Ω. There has been considerable work on obtaining the expected number of real zeros of algebraic𝑃𝑛(𝑥,𝜔)≡𝑃𝑛(𝑥)=𝑛𝑗=0ğ‘Žğ‘—(𝜔)𝑥𝑗,(1.1)and trigonometric ∑𝑛𝑗=0ğ‘Žğ‘—(𝜔)cos𝑗𝜃 polynomials with random coefficients ğ‘Žğ‘—(𝜔)s. The study of the random algebraic polynomials was initiated by Kac [1], and the recent works include [2, 3]. It is shown that under general assumptions for the distribution of coefficients the expected number of real zeros is asymptotic to (2/𝜋)log𝑛 as ğ‘›â†’âˆž. For the case of random trigonometric polynomials, Dunnage [4] obtained the first result which was later generalized by Wilkins Jr. [5, 6] and recently studied in [7, 8]. It is shown that, again for a wide class of distributions for the coefficients, there are significantly more real zeros in the case of trigonometric polynomial compared with the algebraic case. The asymptotic value for the expected number of zeros for the latter case is √2𝑛/3. Besides the comprehensive book of Bharucha-Reid and Sambandham [9], the earlier results of general topics on random polynomials are reviewed in [10].

Motivated by the interesting work of Edelman and Kostlan [11], who, among others, considered polynomials of the form ∑𝑛𝑗=0ğ‘Žğ‘—(𝜔)(𝑛𝑗)1/2𝑥𝑗, [2, 12] obtained many characteristics, like the number of real zeros or the number of maxima of these types of polynomials. This is interesting as they showed that for this case of nonidentically distributed coefficients the expected number of real zeros is √𝑂(𝑛), which is significantly more than the classical algebraic case but less than that of trigonometric polynomials. Also in this direction of nonidentical coefficients, a case in which the mean of coefficients ğ‘Žğ‘—(𝜔) increases with 𝑗 is studied in [3, 13]. Now it would be interesting to study a random polynomial formed by combining the above two distribution laws. It is natural to ask, for instance, what would be the behavior of 𝑃𝑛(𝑥) in (1.1) if for constants 𝜇 and ğœŽ the mean and variance of coefficients are 𝐸(ğ‘Žğ‘—(𝜔))=(𝑛𝑗)𝜇𝑗+1 and var(ğ‘Žğ‘—(𝜔))=(𝑛𝑗)ğœŽ2𝑗.

With the latter assumption of the distribution of the coefficients, we first show that if 𝜇=0, the expected number of real zeros of 𝑃𝑛(𝑥) denoted by 𝐸𝑁𝑛,𝑃(0,∞)≡𝐸𝑁𝑛(0,∞) is independent of ğœŽ. The case of nonzero 𝜇 is studied in Theorem 1.2. The analysis for the general case is complicated, and we only give the result for a case that 𝜇=ğœŽ2. We prove the following theorem.

Theorem 1.1. For 𝜇=0 and ğœŽ2>0, the expected number of real zeros of 𝑃𝑛(𝑥) is independent of ğœŽ2. That is 𝐸𝑁𝑛(−∞,0)=𝐸𝑁𝑛√(0,∞)=𝑛2.(1.2)

The analysis for the case of 𝜇≠0 would be complicated. Without loss of much generality and certainly interest, we restrict ourselves to the case of 𝜇=ğœŽ2. We prove the following theorem.

Theorem 1.2. The expected number of real zeros of 𝑃𝑛(𝑥) for different values of 𝜇 satisfies ğ¸ğ‘ğ‘›âŽ§âŽªâŽ¨âŽªâŽ©âˆ¼î‚€âˆš(0,∞)=𝑂(1),if𝜇=ğœŽ>1,𝑛22√1−arctan𝜇1−𝜇,if0<𝜇=ğœŽ<1.(1.3)For 𝑥 negative and for every 𝜇=ğœŽ2, 𝐸𝑁𝑛√(−∞,0)∼𝑛2.(1.4)

2. Moments

In order to obtain the expected number of real zeros we use a generalization of the well-known Kac-Rice formula initiated in [1, 14, 15]. To this end, we need the following moments of 𝑃𝑛(𝑥) and its dervative ğ‘ƒî…žğ‘›(𝑥). First, we assume the general assumptions on the means and the variances of coefficients as stated above. That is, 𝐸(ğ‘Žğ‘—)=(𝑛𝑗)𝜇𝑗+1 and var(ğ‘Žğ‘—)=(𝑛𝑗)ğœŽ2𝑗. Since these coefficients are independent, it is easy to show𝛼=𝐸(𝑃𝑛(𝑥))=𝜇𝑛𝑗=0𝑛𝑗(𝑥𝜇)𝑗=𝜇(1+𝜇𝑥)𝑛,(2.1)𝛽=𝐸(ğ‘ƒî…žğ‘›(𝑥))=𝜇2𝑛𝑗=0𝑗𝑛𝑗(𝜇𝑥)𝑗−1=𝑛𝜇2(1+𝜇𝑥)𝑛−1,𝐴(2.2)2=var(𝑃𝑛(𝑥))=𝑛𝑗=0𝑛𝑗(ğœŽğ‘¥)2𝑗=(1+ğœŽ2𝑥2)𝑛,𝐵(2.3)2=var(ğ‘ƒî…žğ‘›(𝑥))=ğœŽ2𝑛𝑗=0𝑗2𝑛𝑗(ğœŽğ‘¥)2=ğ‘›ğœŽ2(1+ğœŽ2𝑥2)𝑛−2(1+ğ‘›ğœŽ2𝑥2),(2.4) and finally𝐶=cov(𝑃𝑛(𝑥),ğ‘ƒî…žğ‘›(𝑥))=ğœŽ2𝑥𝑛𝑗=0𝑗𝑛𝑗(ğœŽğ‘¥)2𝑗−2=ğ‘›ğœŽ2𝑥(1+ğœŽ2𝑥2)𝑛−1.(2.5) Then from (2.3)–(2.5) we can obtainΔ2=𝐴2𝐵2−𝐶2=ğ‘›ğœŽ2(1+ğœŽ2𝑥2)2𝑛−2.(2.6)With the above notations, we can now write the Kac-Rice for the expected number of real zeros of 𝑃𝑛(𝑥) in the interval (ğ‘Ž,𝑏) as, see also [10, page 43],𝐸𝑁𝑛(ğ‘Ž,𝑏)=𝐼1(ğ‘Ž,𝑏)+𝐼2(ğ‘Ž,𝑏),(2.7)where𝐼1(ğ‘Ž,𝑏)=ğ‘ğ‘ŽÎ”ğœ‹ğ´2−𝛼exp2𝐵2+𝛽2𝐴2−2𝛼𝛽𝐶2Δ2𝐼𝑑𝑥,(2.8)2(ğ‘Ž,𝑏)=ğ‘ğ‘Žâˆš2|𝛽𝐴2−𝐶𝛼|𝜋𝐴3−𝛼exp22𝐴2erf|𝛽𝐴2−𝐶𝛼|√2𝐴Δ𝑑𝑥,(2.9) where as usual ∫erf(𝑥)=𝑥0exp(−𝑡2)𝑑𝑡. Now we can progress and evaluate further the following terms required in the Kac-Rice formulae (2.7)–(2.9). From (2.1)–(2.5) we can derive

𝛼2𝐵2+𝛽2𝐴2−2𝛼𝛽𝐶=𝑛𝜇2(1+𝜇𝑥)2𝑛−2(1+ğœŽ2𝑥2)𝑛−2(ğ‘›ğœŽ4𝑥2+ğœŽ2+2ğœ‡ğ‘¥ğœŽ2+𝜇2𝑥2ğœŽ2+𝑛𝜇2−2ğ‘›ğœ‡ğœŽ2𝑥).(2.10)This together with (2.6) yields𝛼2𝐵2+𝛽2𝐴2−2𝛼𝛽𝐶2Δ2=𝜇2(1+𝜇𝑥)2𝑛−2{𝑛(ğœŽ2𝑥−𝜇)2+ğœŽ2(1+𝜇𝑥)2}2ğœŽ2(1+ğœŽ2𝑥2)𝑛.(2.11)

3. Proof of Theorems

First in the case of 𝜇=0 from (2.7) and (2.3)–(2.6) by letting 𝑦=ğœŽğ‘¥ we can show𝐸𝑁𝑛√(0,∞)=ğ‘›ğœ‹î€œâˆž0ğœŽ1+ğœŽ2𝑥2=âˆšğ‘‘ğ‘¥ğ‘›ğœ‹î€œâˆž0𝑑𝑦1+𝑦2=√𝑛2.(3.1)This proves Theorem 1.1. Now we proceed with the more general case of 𝜇≠0. As explained above, in order to simplify the analysis we let 𝜇=ğœŽ2. This yields (2.11) to𝛼2𝐵2+𝛽2𝐴2−2𝛼𝛽𝐶2Δ2=𝑓𝑛(𝑥,𝜇)𝑔𝑛(𝑥,𝜇),(3.2)where𝑔𝑛(𝑥,𝜇)=𝑛𝜇2(𝑥−1)2+𝜇(1+𝜇𝑥)2,(3.3)and for all sufficiently large 𝑛,𝑓𝑛(𝑥,𝜇)=(1+𝜇𝑥)2𝑛−2(1+𝜇𝑥2)𝑛∼(1+𝜇𝑥)2𝑛(1+𝜇𝑥2)𝑛=1+𝜇2𝑥2+2𝜇𝑥1+𝜇𝑥2𝑛.(3.4)Now we assume 𝑥>0. Then if we let 𝜇>1, since1+𝜇2𝑥2+2𝜇𝑥1+𝜇𝑥2>1,(3.5)we can see that 𝑓𝑛(𝑥,𝜇)→∞ as ğ‘›â†’âˆž. Therefore the exponential term that appears in (2.8) tends to zero exponentially fast. Hence the only contribution to 𝐸𝑁𝑛(0,∞) is from 𝐼2(0,∞). In the following, we show that the latter is 𝑂(1). To this end, we note that since from the definition for all 𝑥, √erf(𝑥)≤𝜋/2, then𝐼21(0,∞)<√2ğœ‹âˆž0𝛽𝐴2−𝐶𝛼𝐴3−𝛼exp22𝐴2𝑑𝑥.(3.6)Now we let 𝑢=𝛼/𝐴, and since (𝑑/𝑑𝑥)(𝛼/𝐴)=(𝛽𝐴2−𝛼𝐶)/𝐴3 from (3.6) we obtain𝐼21(0,∞)<√2ğœ‹âˆž0−𝑢exp221𝑑𝑢≤2.(3.7)This completes the first part of Theorem 1.2. On the other hand, if 𝜇<1, the behavior of 𝑓𝑛(𝑥,𝜇) will depend on 𝑥. That is for 0<𝑥<2/(1−𝜇), 𝑓𝑛(𝑥,𝜇)→∞ as ğ‘›â†’âˆž and for 𝑥>2/(1−𝜇), 𝑓𝑛(𝑥,𝜇)→0 as ğ‘›â†’âˆž. Therefore the only contribution to 𝐸𝑁𝑛(0,∞) from 𝐼1 is in the interval (2/(1−𝜇),∞) as 𝐼1(0,2/(1−𝜇)) will tend to zero exponentially fast. Also for √𝜈=𝜇𝑥,𝐼12∼√1−𝜇,âˆžğ‘›ğœ‹î€œâˆž2/(1−𝜇)√𝜇1+𝜇𝑥2√𝑑𝑥=ğ‘›ğœ‹î€œâˆž2/(1−𝜇)𝑑𝜈1+𝜈2∼√𝑛22√1−arctan𝜇.1−𝜇(3.8)The above argument for 𝐼2(0,∞) in (3.7) remains valid, and therefore we have proof of the first part of Theorem 1.2.

For 𝑥<0 without loss of generality, we only consider the case of 𝜇>0 (since 𝜇=ğœŽ2). For this case 𝑔𝑛(𝑥,𝜇) remains positive. However, for 𝑥2>𝜖/𝜇, where for ğ‘Ž=1−loglog𝑛10/log𝑛 we let 𝜖=ğ‘›âˆ’ğ‘Ž, we have, (see also [10, page 31]),(1+𝜇𝑥2)𝑛>(1+𝜖)𝑛{(1+ğ‘›âˆ’ğ‘Ž)ğ‘›ğ‘Ž}𝑛1âˆ’ğ‘Ž=exp(𝑛1âˆ’ğ‘Ž)∼𝑛10.(3.9)Hence𝑓𝑛(𝑥,𝜇)<(1+𝜇𝑥2)−𝑛<𝑛−10,(3.10)which tends to zero very fast as ğ‘›â†’âˆž. Therefore the exponential term in 𝐼1 tends to be one, and hence𝐼1𝜖𝜇∼,âˆžğ‘›ğœ‹î€œâˆžâˆšğœ–/𝜇𝑑𝑥1+𝑥2∼√𝑛2.(3.11)Also in the interval √(0,𝜖/𝜇),𝐼10,𝜖𝜇<√0𝜖/𝜇Δ𝜋𝐴2√𝑑𝑥<𝑛𝜇𝜋√0𝜖/𝜇𝑑𝑥1+𝜇𝑥2∼√𝑛2𝜋arctan𝜖,(3.12)which is small. This completes the proof of Theorem 1.2.

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Copyright © 2009 K. Farahmand and M. Sambandham. 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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