Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

Abstract

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form , where is a sequence of standard Gaussian random variables, is . It is shown that the asymptotic value of expected number of times the polynomial crosses the level is also as long as does not exceed , where . The number of oscillations of about will be less than asymptotically only if , where or . In the former case the number of oscillations continues to be a fraction of and decreases with the increase in value of . In the latter case, the number of oscillations reduces to and almost no trace of the curve is expected to be present above the level if log .

1. Introduction

Let be a fixed probability space and let be a sequence of independent random variables defined on . The sum is traditionally known as a random algebraic polynomial if , a random trigonometric polynomial if or , and a random hyperbolic polynomial if or . 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 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 where 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 . Farahmand [3] calculated the asymptotic estimate of oscillations of about if . Some of the other works in this direction are due to Mahanti [4–6]. Wilkins [7] determined real zeros of when , . We observe that the asymptotic value of the oscillations of random hyperbolic polynomials is in each of these cases.

One is tempted to ask whether has more than 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 . Therefore, we examine what effect this new assumption on variance of the coefficients has on number of oscillations of . In other words, we calculate the number of oscillations of the polynomial where are normally distributed random variables with mean zero and variance one.

In Theorem 1 we have shown that the number of real zeros of is substantially larger than that of . Moreover, there is a significant difference in the way real zeros of and lie on the -axis. Most of the real zeros of are confined to the interval and there are negligible numbers of them if (see [4]). But there are large numbers of real zeros of outside . This phenomenon can be deduced from the formula given in (26) and Lemma 10. In fact, the number of real zeros of in the region ,  , is dependent on . The real zeros decrease in number with increase in in the region and there are negligible numbers of them only when .

Let be any function of such that . In Theorem 2 we have shown that the number of oscillations of about the line , where , is equal to its axis crossings. Thus, for all these values of one can say that most of the oscillations of that cross the -axis reach up to the level . If or , we have proved in the theorem that the asymptotic value of number of oscillations of about is less than . If the number of oscillations continues to be a fraction of and decreases with the increase in value of . If the number of oscillations is reduced to . Inequality (35) together with Lemma 14 provides a glimpse of the manner in which the number of oscillations decreases with increase in value of . Inequality (35) also shows that there is hardly any trace of the curve above the level if .

Let the coefficients of be standard normal random variables. The expected number of oscillations of about the line , , has been denoted by us as in the following two theorems.

Theorem 1. For sufficiently large ,

Theorem 2. For sufficiently large ,(i) if ,(ii) if ,(iii) if , , and ,(iv) if and ,(v) if and .

Two more differences in behaviour of and are noteworthy. In what follows we will find out that most of the axis crossings of reach the level . However, the branches of that cross the axis do not travel beyond , where (see Mahanti and Sahoo [6]). Almost all of the polynomial lie below the level , where (Mahanti and Sahoo [6]). However, a large part of 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 in the interval we can show that where , ,  , , ,   and  . Since the coefficients of are independent and , it is easy to derive using little algebra that Let Formula (4) now can be written as where As a special case, we can also obtain the famous Kac-Rice formula [10] for expected number of zeros of by putting in (9). Thus we have

3. Preliminary Analysis

To evaluate the integrals in (9) and (11) we need to find out the dominant terms of , , and . The inequalities mentioned in Lemmas 4–7 will be helpful for the purpose. We first mention the following form of L’Hôpital’s Rule which we use to derive some of the inequalities.

Lemma 3 (the monotonic form of L’Hôpital’s Rule [11]). For , let be continuous on and differentiable on with on . If is increasing or decreasing on , then so are and . If is strictly monotone, then the monotonicity of the above two quotients is also strict.

Lemma 4. If , then . In particular in the interval Thus if .

Proof. The series representation of is given by [12, page 42] , where . Using the fact that [12, page 1038], we have .
Now, is a monotonically decreasing function of . Therefore, the power series of converges absolutely and uniformly for . Integrating term by term we have the alternative series for . Since the coefficients of are monotonically decreasing, we find that the first part of the lemma is true.
Let , where and , with . Then It is easy to verify that if . Using the series representation of we find that Therefore if . As a consequence, is strictly decreasing in the interval . By Lemma 3, is decreasing in and by L’Hôpital’s Rule . Therefore, . Let , where and with . Then , where and with . Now , which is increasing. So, by Lemma 3, is strictly increasing. By L’Hôpital’s Rule we find that . Therefore,

Lemma 5. If , then .

Proof. Let , , and . Observe that . Let and . Then is a monotonically decreasing and is monotonically increasing function of . By L’Hôpital Rule . Therefore, by Lemma 3, and for .

Lemma 6. Let ,  . Then is a monotonically decreasing function of in if and is a monotonically decreasing function of in if and .

Proof. Since to prove the lemma we show that under the conditions mentioned in the lemma. We first note that . Therefore, if and , since its value is zero at .
Now let . Let , where and with ,  . As is a decreasing function of , it follows from Lemma 3 that is monotonically decreasing. Therefore, for a fixed , increases with . As and , there exists an integer such that, for , . In other words, for , at . Since , we conclude that if and . We can find out in the following manner.
We note that as and is zero at . So if ; that is, .

Lemma 7. If , then

Proof. Let and . Let and . We observe that . After differentiation we have We now use the following power series representation (see [12]) convergent for , , where is the Bernoulli number of degree .
Then, We observe that is nonnegative in since it vanishes at zero and its derivative, that is, is positive.
It follows from (19) that ( is a nondecreasing function in since and by L’Hôpital’s Rule , we obtain the proof of Lemma 7 using Lemma 3.

Lemma 8. Consider where

Proof. Consider the following power series representation of [12, eq. 3.321.1] . The power series converges absolutely and uniformly for . Hence [12, page 346] where . It is easy to see that is nonincreasing and as . Thus, the statement of the lemma is true.

Lemma 9. Let ; then

Proof. By Lemma 4, defined in (8) can be written as By Lemma 6 we find that in (0,1) and if . Also, if and . So, ,   . Also using Lemma 4 and (5) we find that . By (11), we see that Lemma 9 is valid.

Lemma 10. If , then

Proof. Using Lemmas 6 and 4 we obtain from (8) that and . The conclusion of the lemma follows now from (11).

4. Proof of Theorem 1

If is taken as we can derive from Lemmas 9 and 10 the following relations: Since the integrand in (11) is an even function of , we have Using Lemma 5 to approximate , we see that Theorem 1 is true.

5. Proof of Theorem 2

To determine we need to calculate and for different ranges of values of as both quantities depend on magnitude of . In Lemma 14 we have calculated . Value of for three different ranges of value of has been calculated in Lemmas 11–13. The relations (32), (34), and (35) and Lemma 14 establish Theorem 2. Note that we only need to calculate and since and are even functions of .

Since , by Lemma 9 we have In order to calculate , we need the dominant term of , which can be calculated with the help of Lemmas 4 and 6 as For brevity, we have written in the following lemmas.

Lemma 11. As long as does not exceed , where ,

Proof. Let ,  . Then by Lemma 9 we find that . Using (30) and Lemma 10, we obtain that . Hence .
Now let ,  , and . It follows from (30) that the maximum value of in is .
Hence . As the integrand of is bounded, we have . By Lemma 10 and (30) we find that . Taking into account (29) we see that (31) is true.

Lemma 12. Let , where either or , but ; then

Proof. By Lemma 10 and (30) we find that the following relations are true: These relations and (29) yield (32).

Lemma 13. Let , where or . Then For large values of where .
Moreover, if and if .

Proof. Let and be the points in where assumes the values and , respectively. It follows from (30) and Lemma 10 that Let and .
Then . Therefore, by (30) and the definition of , we have and . We obtain (34) if (29) is also considered. Inequality (35) follows immediately from Lemma 7. From this inequality the other two estimates of valid for and such that follow.

Lemma 14. The dependence of on is given by the following relations:(i)if , then ,(ii)if , then ,(iii)if , then , where .

Proof. Using Lemma 4 we find that in Hence, , where .
On the other hand, using Lemma 6 and the definitions of and in , where , we have .
Let . It is not difficult to see that in the interval where .
Clearly if . The other two parts of the lemma are found to be true by virtue of Lemma 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.