Random Trigonometric Polynomials with Nonidentically Distributed Coefficients

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of a0 a1 cos θ a2 cos 2θ · · · an cosnθ and a0 a1 cos θ b1 sin θ a2 cos 2θ b2 sin 2θ · · · an cosnθ bn sinnθ,we give a closed form for the above expected value. With some mild assumptions on the coefficients we allow the means and variances of the coefficients to differ from each others. A case of reciprocal random polynomials for both above cases is studied.


Introduction
There are mainly two different forms of random trigonometric polynomial previously studied.They are T θ n j 0 a j cos jθ, D θ n j 0 a j cos jθ b j sin jθ .

1.1
Dunnage 1 first studied the classical random trigonometric polynomial T θ .He showed that in the case of identically and normally distributed coefficients {a j } n j 0 with μ j ≡ 0 and σ 2 j ≡ 1, j 0, 1, 2, . . ., n, the number of real zeros in the interval 0, 2π , outside of an exceptional set of measure zero, is 2n/ √ 3 O{n 11/13 log n 3/13 }, when n is large.Subsequent papers mostly assumed an identical distribution for the coefficients and obtained 2n/ √ 3 as the asymptotic formula for the expected number of real zeros.In 2-4 it is International Journal of Stochastic Analysis shown that this asymptotic formula remains valid when the expected number of real zeros of the equation T θ K, known as K-level crossing, is considered.The work of Sambandham and Renganathan 5 and Farahmand 6 among others obtained this result for different assumptions on the distribution of the coefficients.Earlier works on random polynomials have been reviewed in Bharucha-Reid and Sambandham 7 , which includes a comprehensive reference.
Later Farahmand and Sambandham 8 study a case of coefficients with different means and variances, which shows an interesting result for the expected number of level crossings in the interval 0, 2π .Based on this work, we study the following two cases in order to better understand how the behavior of random trigonometric polynomials is affected by the different assumptions of the distribution on the coefficients for both T θ and D θ , defined above.
To this end we allow all the coefficients to have different means and variances.Also, motivated by the recent developments on random reciprocal polynomials, we assume the coefficients a j and a n−j have the same distribution.In 9 for the case of random algebraic polynomial a j ≡ a n−j is assumed.Further in order to overcome the analysis we have to make the following assumptions on the means and variances.Let max{σ 1.2 We study the case of D θ in Theorem 3.1 later.We first give some necessary identities.

Preliminary Analysis
In order to be able to prove the theorem, we need to define some auxiliary results.Let

2.1
Then from Farahmand 10, page 43 , we have the extension of the Kac-Rice formula for our case as where

2.3
As usual, erf Now we are going to define the following functions to make the estimations.At first, we define S θ sin 2n 1 θ/ sin θ and to be continuous at θ jπ see also 10, page 74 .Let be any positive value arbitrary at this point, to be defined later.Since for θ

2.6
Now using the above identities and by expanding sin θ 1 2 n j 1 cos 2jθ , we can show

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In a similar way to 10 , we define Furthermore, we have

2.9
Now using these identities for Q θ , Q θ , and Q θ and by expanding 2 sin θ n j 1 sin 2jθ, we can get a series of the following results:

2.10
Now we are in position to give a proof for Theorem 1.1 for T θ in the intervals , π − and π , 2π − .In order to avoid duplication the remaining intervals for both cases of T θ and D θ are discussed together later.

The Proof
Case 1.Here we study the random trigonometric polynomial in the classical form of T θ n j 0 a j cos jθ as assumed in Theorem 1.1 and prove the theorem in this section.To this end, we have to get all the terms in the Kac-Rice formula, such as A 2 , B 2 , C, α, and β.Since the property σ 2 j σ 2 n−j , using the results obtained in Section 2 of 2.7 and 2.10 , we can have all the terms needed to calculate formula 2.2 .At first, we get the variance of the polynomial, that is,

3.1
Next, we calculate the variance of its derivative T θ with respect to θ:

3.2
At last, it turns to the covariance between the polynomial and its derivative

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It is also easy to obtain the means of T θ and its derivative as

3.5
From 2.3 and the results of 3.1 -3.5 , we therefore have

3.6
Now before considering the zeros in the small interval of length we consider the polynomial D θ .

3.7
Similarly, using the same results obtained from Section 2, we can get the following terms.At first, we get the means of the polynomial and its derivative separately

3.8
International Journal of Stochastic Analysis 7 Then we obtain the variance of the polynomial 3.9 Next, we calculate the variance of its derivative with respect to θ:

3.10
At last, it turns to the covariance between the polynomial and its derivative:

3.12
International Journal of Stochastic Analysis From 2.3 and 3.8 -3.12 , we therefore have

3.13
This is the main contribution to the number of real zeros.In the following we show there is a negligible number of zeros in the remaining intervals of length .For the number of real roots in the interval 0, ε , 2π − , 2π or π − ε, π ε , we use Jensen's theorem 11, page 300 .The method we used here is applicable to both of the cases we discussed above.Here we take the first case as the example to prove that the roots of these intervals are negligible.Let m n j 0 μ j and s 2 n j 0 σ 2 j .As T 0 is normally distributed with mean m and variance s 2 , for any constant k

3.14
Also since | cos 2εe iθ | ≤ 2e 2nε we have Pr N ε ≥ j ≤ 3nε and δ is any positive number, so the error terms become small and we can prove the theorem.

2n j 3nε 1 Pr
N ε > j ≤ 3nε d n 1−k O nε , Therefore, except for sample functions in an ω-set of measures not exceeding dn − k, T 2εe iθ ≤ 3n 1 2k exp 2nε .samplefunctions in an ω-set of measure not exceeding dn −k 2/n k √ πs2.This implies that we can find an absolute constant d such thatPr N ε > 2nε 1 2k log n log 3 ≤ n −k d 2 s 2 √ π ≤ d n −k .3.19Let 3nε be the greatest integer less than or equal to 3nε.Then since the number of real zeros of D θ is at most 2n we have