Random Trigonometric Polynomials with Nonidentically Distributed Coefficients

Abstract

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+𝑎₁cos𝜃+𝑎₂cos2𝜃+⋯+𝑎_𝑛cos𝑛𝜃 and 𝑎₀+𝑎₁cos𝜃+𝑏₁sin𝜃+𝑎₂cos2𝜃+𝑏₂sin2𝜃+⋯+𝑎_𝑛cos𝑛𝜃+𝑏_𝑛sin𝑛𝜃, 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.

1. Introduction

There are mainly two different forms of random trigonometric polynomial previously studied. They are

𝑇(𝜃)=^𝑛_𝑗=0𝑎_𝑗cos𝑗𝜃,𝐷(𝜃)=^𝑛_𝑗=0(𝑎_𝑗cos𝑗𝜃+𝑏_𝑗sin𝑗𝜃). (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 𝜎²_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(log𝑛)^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 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{𝜎²_𝑗}=𝜎^∗_𝑛²=𝑂(𝑛^2/3) and also |𝜎²_𝑗−𝜎²_𝑗+1|≤𝜎^∗_𝑛²/𝑛. For the means, we assume max{|𝜇_𝑗|}=𝜇^∗_𝑛=𝑂(√𝑛) and |𝜇_𝑗−𝜇_𝑗+2|≤𝜇^∗_𝑛/𝑛. We also need 𝜎_𝑛^∗2,𝜎²_∗𝑛, where 𝜎²_∗𝑛=min{𝜎²_𝑗} and 𝜇^∗_𝑛 is chosen such that for any positive constant 𝛿, (𝜎_𝑛^∗3𝑛^𝛿−1/𝜎³_∗𝑛)→0 and (𝜇^∗_𝑛𝑛^𝛿−1/2/𝜎_∗𝑛)→0 as 𝑛→∞. Then for 𝜎²_∗𝑛 finite, we have the following theorem.

Theorem 1.1. If the coefficients 𝑎_𝑗, 𝑗=1,…,𝑛 of 𝑇(𝜃)=∑^𝑛_𝑗=0𝑎_𝑗cos𝑗𝜃 are normally distributed with mean 𝜇_𝑗 and variance 𝜎²_𝑗, where 𝜎²_𝑗=𝜎²_𝑛−𝑗, then the mathematical expectation of the number of real zeros of the 𝑇(𝜃) satisfies 𝐸𝑁_𝑇(0,2𝜋)=2∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗(𝑛²/2−𝑛𝑗+𝑗²)∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗. (1.2)

We study the case of 𝐷(𝜃) in Theorem 3.1 later. We first give some necessary identities.

2. Preliminary Analysis

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

𝐴²=var{𝑇(𝜃)},𝐵²=var{𝑇^′(𝜃)},𝐶=cov{𝑇(𝜃),𝑇^′(𝜃)},Δ=𝐴²𝐵²−𝐶²,𝛼=𝐸{𝑇(𝜃)},𝛽=𝐸{𝑇^′(𝜃)}. (2.1) Then from Farahmand [10, page 43], we have the extension of the Kac-Rice formula for our case as

𝐸𝑁(𝑎,𝑏)=𝐼₁(𝑎,𝑏)+𝐼₂(𝑎,𝑏), (2.2) where

𝐼₁(𝑎,𝑏)=^𝑏_𝑎Δ𝜋𝐴²𝛼exp(−²𝐵²+𝛽²𝐴²−2𝛼𝛽𝐶2Δ²𝐼)𝑑𝜃,₂(𝑎,𝑏)=^𝑏_𝑎√2|𝛽𝐴²−𝐶𝛼|𝜋𝐴³𝛼exp(−²2𝐴²)erf(|𝛽𝐴²−𝐶𝛼|√2𝐴Δ)𝑑𝜃. (2.3) As usual, erf(|𝛽𝐴²−𝐶𝛼|/√2𝐴Δ) is the function defined as

erf(𝑥)=^𝑥₀exp(−𝑡²)𝑑𝑡=√𝜋Φ(𝑥√√2)−𝜋2. (2.4) Now we are going to define the following functions to make the estimations. At first, we define 𝑆(𝜃)=sin(2𝑛+1)𝜃/sin𝜃 and to be continuous at 𝜃=𝑗𝜋 see also [10, page 74]. Let 𝜖 be any positive value arbitrary at this point, to be defined later. Since for 𝜃∈(𝜀,𝜋−𝜀),(𝜋+𝜀,2𝜋−𝜀), we have |𝑆(𝜃)|<1/sin𝜀, we can obtain

𝑆(𝜃)=𝑂(1/𝜀). (2.5) Furthermore,

𝑆^′(𝜃)=(2𝑛+1)cos(2𝑛+1)𝜃𝑛sin𝜃−cot𝜃𝑆(𝜃)=𝑂(𝜀𝑆),_′′(𝜃)=−(2𝑛+1)²sin(2𝑛+1)𝜃−sin𝜃(2𝑛+1)cos𝜃cos(2𝑛+1)𝜃sin²𝜃−cot𝜃𝑆^′(𝜃)+csc²𝑛𝜃𝑆(𝜃)=𝑂(²𝜀). (2.6)

Now using the above identities and by expanding sin𝜃(1+2∑^𝑛_𝑗=1cos2𝑗𝜃), we can show

^𝑛_𝑗=1cos2𝑗𝜃=sin(2𝑛+1)𝜃−12sin𝜃2=𝑆(𝜃)−121=𝑂(𝜀),_𝑛𝑗=11𝑗sin2𝑗𝜃=−2{^𝑛_𝑗=1cos2𝑗𝜃}^′1=−4𝑆^′𝑛(𝜃)=𝑂(𝜀),_𝑛𝑗=1𝑗²1cos2𝑗𝜃=−4{^𝑛_𝑗=1cos2𝑗𝜃}^′′1=−8𝑆^′′𝑛(𝜃)=𝑂(²𝜀). (2.7) In a similar way to [10], we define 𝑄(𝜃)=cos𝜃−cos(2𝑛+1)𝜃/2sin𝜃, then for 𝜃∈(𝜀,𝜋−𝜀),(𝜋+𝜀,2𝜋−𝜀), since cos𝜃−cos(2𝑛+1)𝜃=2sin(𝑛+1)𝜃sin𝑛𝜃, we have |𝑄(𝜃)|<1/sin𝜀. Hence, we can obtain

1𝑄(𝜃)=𝑂(𝜀). (2.8) Furthermore, we have 𝑄^′(𝜃)=−sin𝜃+(2𝑛+1)sin(2𝑛+1)𝜃𝑛2sin𝜃−cot𝜃𝑄(𝜃)=𝑂(𝜀𝑄),_′′(𝜃)=−cos𝜃+(2𝑛+1)²cos(2𝑛+1)𝜃+2sin𝜃cos𝜃[sin𝜃−(2𝑛+1)sin(2𝑛+1)𝜃]2sin²𝜃−cot𝜃𝑄^′(𝜃)+csc²𝑛𝜃𝑄(𝜃)=𝑂(²𝜀). (2.9) Now using these identities for 𝑄(𝜃), 𝑄^′(𝜃), and 𝑄^′′(𝜃) and by expanding 2sin𝜃∑^𝑛_𝑗=1sin2𝑗𝜃, we can get a series of the following results:

^𝑛_𝑗=1sin2𝑗𝜃=cos𝜃−cos(2𝑛+1)𝜃12sin𝜃=𝑄(𝜃)=𝑂(𝜀),_𝑛𝑗=11𝑗cos2𝑗𝜃=2{^𝑛_𝑗=1sin2𝑗𝜃}^′=12𝑄^′𝑛(𝜃)=𝑂(𝜀),_𝑛𝑗=1𝑗²1sin2𝑗𝜃=−4{^𝑛_𝑗=1sin2𝑗𝜃}^′′1=−4𝑄^′′𝑛(𝜃)=𝑂(²𝜀). (2.10) Now we are in position to give a proof for Theorem 1.1 for 𝑇(𝜃) in the intervals (𝜖,𝜋−𝜖) and (𝜋+𝜖,2𝜋−𝜖). In order to avoid duplication the remaining intervals for both cases of 𝑇(𝜃) and 𝐷(𝜃) are discussed together later.

3. The Proof

Case 1. Here we study the random trigonometric polynomial in the classical form of 𝑇(𝜃)=∑^𝑛_𝑗=0𝑎_𝑗cos𝑗𝜃 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 𝐴²,𝐵²,𝐶,𝛼,and𝛽. Since the property 𝜎²_𝑗=𝜎²_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, 𝐴²_𝑇=^(𝑛−1)/2_𝑗=0var{𝑎_𝑗}{cos²𝑗𝜃+cos²=(𝑛−𝑗)𝜃}_{(𝑛−1)/2𝑗=0}𝜎²_𝑗+cos𝑛𝜃^(𝑛−1)/2_𝑗=0𝜎²_𝑗={cos(𝑛−2𝑗)𝜃}_{(𝑛−1)/2𝑗=0}𝜎²_𝑗𝜎+𝑂(^𝑛∗2𝜀). (3.1) Next, we calculate the variance of its derivative 𝑇^′(𝜃) with respect to 𝜃: 𝐵²_𝑇=^(𝑛−1)/2_𝑗=0var{𝑎_𝑗}{𝑗²sin²𝑗𝜃+(𝑛−𝑗)²sin²=(𝑛−𝑗)𝜃}_{(𝑛−1)/2𝑗=0}𝜎²_𝑗(𝑛²2−𝑛𝑗+𝑗²)−^(𝑛−1)/2_𝑗=0𝜎²_𝑗𝑗²2−cos2𝑗𝜃_{(𝑛−1)/2𝑗=0}𝜎²_𝑗(𝑛²2−𝑛𝑗+𝑗²=)cos2(𝑛−𝑗)𝜃_{(𝑛−1)/2𝑗=0}𝜎²_𝑗(𝑛²2−𝑛𝑗+𝑗²𝑛)+𝑂(²𝜎_𝑛^∗2𝜀). (3.2) At last, it turns to the covariance between the polynomial and its derivative 𝐶_𝑇=^(𝑛−1)/2_𝑗=0−𝑗𝜎²_𝑗sin𝑗𝜃cos𝑗𝜃=−^(𝑛−1)/2_𝑗=0𝑗2𝜎²_𝑗𝑛{sin2𝑗𝜃−sin2(𝑛−𝑗)𝜃}−2^(𝑛−1)/2_𝑗=0𝜎²_𝑗sin2(𝑛−𝑗)𝜃=𝑂(𝑛𝜎_𝑛^∗2𝜀). (3.3) Then, from (3.1), (3.2), and (3.3), we can get Δ²_𝑇=𝐴²_𝑇𝐵²_𝑇−𝐶²_𝑇=^(𝑛−1)/2_𝑗=0𝜎²_𝑗^(𝑛−1)/2_𝑗=0𝜎²_𝑗(𝑛²2−𝑛𝑗+𝑗²𝑛)+𝑂(²𝜎_𝑛^∗2𝜀). (3.4) It is also easy to obtain the means of 𝑇(𝜃) and its derivative as 𝛼_𝑇=∑^𝑛_𝑗=0𝜇_𝑗𝜇cos𝑗𝜃=𝑂(^∗𝑛𝜀𝛽),_𝑇=−∑^𝑛_𝑗=0𝑗𝜇_𝑗sin𝑗𝜃=𝑂(𝑛𝜇^∗_𝑛𝜀). (3.5) From (2.3) and the results of (3.1)–(3.5), we therefore have 𝐸𝑁_𝑇(𝜀,𝜋−𝜀)∼_𝜀^𝜋−𝜀Δ𝜋𝐴²∼𝑑𝜃_{𝜀𝜋−𝜀}∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗(𝑛²/2−𝑛𝑗+𝑗²)𝜋²∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗=𝑑𝜃∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗(𝑛²/2−𝑛𝑗+𝑗²)∑^(𝑛−1)/2_𝑗=0𝜎²_𝑗. (3.6) Now before considering the zeros in the small interval of length 𝜖 we consider the polynomial 𝐷(𝜃).

Case 2. We have to make the assumptions a little different. In this case let max|{𝜎²_𝑎𝑗}−{𝜎²_𝑏𝑗}|=𝜎^∗_𝑛² and |𝜎²_𝑎𝑗+𝜎²_𝑏𝑗−𝜎²_𝑎𝑗+1−𝜎²_𝑏𝑗+1|≤𝜎^∗_𝑛²/𝑛. For the means, we assume max{𝜇_𝑎𝑗}=𝜇^∗_𝑎𝑛 and |𝜇_𝑎𝑗−𝜇_𝑎𝑗+2|≤𝜇^∗_𝑎𝑛/𝑛,max{𝜇_𝑏𝑗}=𝜇^∗_𝑏𝑛 and |𝜇_𝑏𝑗−𝜇_𝑏𝑗+2|≤𝜇^∗_𝑏𝑛/𝑛, and max|{𝜇_𝑎𝑗}−{𝜇_𝑏𝑗}|=𝜇^∗_𝑛.

Theorem 3.1. Consider the polynomial 𝐷(𝜃)=∑^𝑛_𝑗=0𝑎_𝑗cos𝑗𝜃+𝑏_𝑗sin𝑗𝜃, where 𝑎_𝑗,𝑏_𝑗 are independent, normally distributed random variables, divided into 𝑛 groups each with its own mean 𝜇_𝑎𝑗,𝜇_𝑏𝑗 and variance 𝜎²_𝑎𝑗,𝜎²_𝑏𝑗, 𝑗=1,2,…,𝑛,. The expected number of real zeros of 𝐷(𝜃) satisfies 𝐸𝑁_𝐷(0,2𝜋)=2∑^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}∑^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}. (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 𝛼_𝐷=∑^𝑛_𝑗=0𝜇_𝑎𝑗cos𝑗𝜃+∑^𝑛_𝑗=0𝜇_𝑏𝑗𝜇sin𝑗𝜃=𝑂(^∗𝑎𝑛+𝜇^∗_𝑏𝑛𝜀𝛽),_𝐷=−∑^𝑛_𝑗=0𝑗𝜇_𝑎𝑗sin𝑗𝜃+∑^𝑛_𝑗=0𝑗𝜇_𝑏𝑗cos𝑗𝜃=𝑂(𝑛𝜇^∗_𝑛𝜀). (3.8) Then we obtain the variance of the polynomial

𝐴²_𝐷=var{^𝑛_𝑗=0𝑎_𝑗cos𝑗𝜃+𝑏_𝑗=1sin𝑗𝜃}2^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗1}+2^𝑛_𝑗=0{𝜎²_𝑎𝑗−𝜎²_𝑏𝑗=1}cos2𝑗𝜃2^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗𝜎}+𝑂(^∗𝑛2𝜀). (3.9) Next, we calculate the variance of its derivative with respect to 𝜃:

𝐵²_𝐷=var{^𝑛_𝑗=0−𝑗𝑎_𝑗sin𝑗𝜃+𝑏_𝑗=1cos𝑗𝜃}2^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗1}+2^𝑛_𝑗=0𝑗²{𝜎²_𝑏𝑗−𝜎²_𝑎𝑗=1}cos2𝑗𝜃2^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗𝑛}+𝑂(²𝜎^∗_𝑛²𝜀). (3.10) At last, it turns to the covariance between the polynomial and its derivative:

𝐶_𝐷=−𝐸{𝑎²_𝑗}𝑗sin𝑗𝜃cos𝑗𝜃+𝐸{𝑏²_𝑗}𝑗sin𝑗𝜃cos𝑗𝜃+𝐸²{𝑎_𝑗}𝑗sin𝑗𝜃cos𝑗𝜃−𝐸²{𝑏_𝑗=}𝑗sin𝑗𝜃cos𝑗𝜃_𝑛𝑗=0−𝑗{𝜎²_𝑏𝑗−𝜎²_𝑎𝑗}sin𝑗𝜃cos𝑗𝜃=𝑂(𝑛𝜎^∗_𝑛²𝜀). (3.11) Then, from (3.9), (3.10), and (3.11), we can get

Δ²_𝐷=14^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗𝑛}+𝑂(𝜀). (3.12) From (2.3) and (3.8)–(3.12), we therefore have

𝐸𝑁_𝐷(𝜖,𝜋−𝜖)∼_𝜀^𝜋−𝜀Δ𝜋𝐴²𝑑𝜃∼_𝜀^𝜋−𝜀∑^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}𝜋²∑^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}=𝑑𝜃∑^𝑛_𝑗=0𝑗²{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}∑^𝑛_𝑗=0{𝜎²_𝑎𝑗+𝜎²_𝑏𝑗}. (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 𝜖. For the number of real roots in the interval (0,𝜀), (2𝜋−𝜖,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=0𝜇_𝑗 and 𝑠²=∑^𝑛_𝑗=0𝜎²_𝑗. As 𝑇(0) is normally distributed with mean 𝑚 and variance 𝑠², for any constant 𝑘

Pr(−𝑛^−𝑘≤𝐷(0)≤𝑛^−𝑘)=(2𝜋𝑠²)^{−1/2𝑛−𝑘}_{−𝑛−𝑘}exp{−(𝑡−𝑚)²2𝑠²<2}𝑑𝑡𝑛^𝑘√𝜋𝑠². (3.14) Also since |cos(2𝜀𝑒^𝑖𝜃)|≤2𝑒^2𝑛𝜀 we have

𝐷(2𝜀𝑒^𝑖𝜃)≤2𝑒^{𝑛2𝑛𝜀}_𝑗=1|𝑎_𝑗|≤2(𝑛+1)𝑒^2𝑛𝜀max_{𝑗=1,…,𝑛}|𝑎_𝑗|. (3.15) Now by Chebyshev's inequality, for any 𝑘>2, we can find a positive constant 𝑑 such that for 0≤𝑗≤𝑛,

Pr(|𝑎_𝑗|≥𝑛^−𝑘)≤𝐸(𝑎²_𝑗)𝑛^2𝑘≤𝑑𝑛^𝑘. (3.16) since 𝜇_𝑛^∗2+𝜎_𝑛^∗2≤𝑛^𝑘. Therefore, except for sample functions in an 𝜔-set of measures not exceeding 𝑑𝑛⁻𝑘,

𝑇(2𝜀𝑒^𝑖𝜃)≤3𝑛^1+2𝑘exp(2𝑛𝜀). (3.17) So we obtain

|𝑇(2𝜀𝑒^𝑖𝜃,𝜔)𝑇(0,𝜔)|≤3𝑛^1+2𝑘exp(2𝑛𝜀), (3.18) except for the sample functions 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 that

Pr{𝑁(𝜀)>2𝑛𝜀+(1+2𝑘)log𝑛+log3}≤𝑛^−𝑘(𝑑+2𝑠²√𝜋)≤𝑑^′𝑛^−𝑘. (3.19) Let [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

𝐸𝑁(𝜀)=^2𝑛_𝑗>0Pr{𝑁(𝜀)≥𝑗}≤3𝑛𝜀+^2𝑛_{𝑗=[3𝑛𝜀]+1}Pr{𝑁(𝜀)>𝑗}≤3𝑛𝜀+𝑑^′𝑛^1−𝑘=𝑂(𝑛𝜀), (3.20) where we choose

𝜇𝜖=max{^∗𝑛𝜎_∗𝑛𝑛^1/2−𝛿,𝜎_𝑛^∗3𝜎³_∗𝑛𝑛^1−𝛿}, (3.21) and 𝛿 is any positive number, so the error terms become small and we can prove the theorem.