Communications on Applied Nonlinear Analysis  

ISSN: 1074-133X 

Vol 31 No. 2s (2024) 

 

 

696 https://internationalpubls.com 

On Mean Convergence of Random Fourier - Hermite Series 

 

Bharatee Mangaraj[1], Sabita Sahoo[2] 

[1]Phd Scholar, Department of Mathematics, Sambalpur University, Odisha, India 

[2] Retired Professor, Department of Mathematics, Sambalpur University, Odisha, India 
[1] mangarajbharatee@suniv.ac.in, [2] sabitamath@suniv.ac.in   

 

Article History: 

Received: 10-04-2024 

Revised: 20-05-2024 

Accepted: 03-06-2024 

Abstract: The work in this article is an initiative to explore random Fourier - Hermite 

series in orthogonal Hermite polynomials. We choose the random coefficients in the 

series to be the Fourier-Hermite coefficients of a symmetric stable process with weight 

function 𝑈(𝑣, 𝑏) = 𝑒
−𝑣2

2 (1 + |𝑣|)𝑏 , where 𝑏 <
1

2
. The existence of these random 

coefficients, which we find to be dependent random variables, is established. The 

random Fourier-Hermite series is proven to be convergent in the sense of mean if the 

scalars in the series are the Fourier-Hermite coefficients of a function 𝑔 in the weighted 

space 𝐿𝑊(𝑣,𝐵)
2 (ℝ), where the weights are given by 𝑊(𝑣, 𝐵) = 𝑒

−𝑣2

2 (1 + |𝑣|)𝐵  with 

𝐵 >
−1

2
 such that 𝑏 < 𝐵. The sum functions of the series is obtained to the stochastic 

integral ∫
∞

−∞
𝑔(𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔). 

2020  MSC.   Primary: 42A38; Secondary: 40A35 

 

Keywords: Symmetric stable process, Stochastic integral, Convergence in mean, 

Convergence in law, Continuity theorem. 

 

1.  Introduction 

Fourier Series in orthogonal functions 𝑒𝑖𝑛𝑢  and other orthogonal polynomials like Hermite 

polynomials, Jacobi polynomials, etc., has a widespread application in physical sciences. Random 

Fourier series(RFS) in orthogonal functions 𝑒𝑖𝑛𝑢 is important in signal processing. For the first time, 

the application of RFS in Hermite polynomial is found in image encryption and decryption in the work 

of Liu and Liu [11] in 2007, who expected its more application in general signal and image processing. 

The RFS they used is an RFT with random coefficients chosen from the unit circle in ℂ randomly. 

This motivated us to explore the random Fourier - Hermite series(RFHS) with different random 

coefficients. Since stable processes are a better model for white noise, the random coefficients 𝐷𝑛(𝜔) 

choosen in this article are Fourier - Hermite coefficients(FHC) of a symmetric stable process(SSP) 

defined as ∫
∞

−∞
𝐻𝑛(𝑢)𝑈(𝑢, 𝑏)  with weights 𝑈(𝑢, 𝑏) = 𝑒

−𝑢2

2 (1 + |𝑢|)𝑏 , 𝑏 <
1

2
. We establish the 

existence of these random variables and demonstrate their dependence. It is proved that the random 

series ∑∞
𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢) in Hermite polynomials 𝐻𝑘(𝑢) convergence in mean to the stochastic 

integral ∫
∞

−∞
𝑔(𝑢)𝑈(𝑢, 𝑏)𝑑𝑋(𝑢, 𝜔)  if the scalars 𝑑𝑘  are FHC of a function 𝑔  in the space 

𝐿𝑊(𝑢,𝐵)
2 (ℝ), defined as 𝑑𝑘: = 𝑟𝑘

2 ∫
∞

−∞
𝑔(𝑢)𝑒−𝑢2

𝐻𝑘(𝑢)𝑑𝑢. 

 

2.  Preliminaries 

Consider 𝜙𝑛(𝑢), 𝑛 ∈ ℕ0, ℕ0: = {0,1,2, . . . } to be a sequence of functions orthonormal concerning a 

measure ℱ(𝑢) that is,  



Communications on Applied Nonlinear Analysis  

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 ∫
𝑏

𝑎
𝜙𝑛(𝑢)𝜙𝑚(𝑢)𝑑ℱ(𝑢) = 𝛿𝑛𝑚, 

where, 𝛿𝑚𝑛 is the Kronecher’s delta function and let  

 𝑔(𝑢) ∼ ∑∞
𝑛=0 𝑎𝑛𝜙𝑛(𝑢) (2.1) 

be the formal expansion of an arbitrary function in terms of this sequence where 𝑎𝑛: =

∫
𝑏

𝑎
𝑔(𝑣)𝜙𝑛(𝑣)𝑑ℱ(𝑣). Many researchers have exhaustively explored the convergence characteristics 

of series of the form (2.1) for a specific set of functions 𝜙𝑛(𝑣). Specifically, the exploration is on the 

following question:  

For what values of 𝑝, 1 ≤ 𝑝 < ∞, does the existence of the integral ∫
𝑏

𝑎
|𝑔(𝑢)|𝑝𝑑ℱ(𝑢) imply,  

 lim
𝑛→∞

∫
𝑏

𝑎
|𝑔(𝑢) − ∑𝑛

𝑘=0 𝑎𝑘𝜙𝑘(𝑢)|𝑝𝑑ℱ(𝑢) = 0? (2.2) 

The sequence 𝜙𝑛(𝑢) forms a basis for the space of these functions when this equation holds for every 

𝑔(𝑢) such that ∫
𝑏

𝑎
|𝑔(𝑢)|𝑝𝑑ℱ(𝑢) exists [25]. If 𝑑ℱ(𝑢) = 𝑊(𝑢)𝑑𝑢, 𝑊(𝑢) is the weight function 

then the sequence 𝜙𝑛(𝑢)(𝑊(𝑢))
1

2 is orthonormal on the classical sense and one led to the formal 

expansion  

 𝑔(𝑢) ∼ ∑∞
𝑛=0 𝑏𝑛𝜙𝑛(𝑢)(𝑊(𝑢))

1

2, 

where 𝑏𝑛: = ∫
𝑏

𝑎
𝑔(𝑣)𝜙𝑛(𝑣)(𝑊(𝑣))

1

2𝑑𝑣. The above question can be read as: for what values of 𝑝, 1 ≤

𝑝 < ∞, does the measurability of 𝑔(𝑢) and the relation ∫
𝑏

𝑎
|𝑔(𝑢)|𝑝𝑑𝑢 < ∞ (i. e., 𝑔 ∈ 𝐿𝑝(ℝ)) imply,  

 lim
𝑛→∞

∫
𝑏

𝑎
|𝑔(𝑢) − ∑𝑛

𝑘=0 𝑏𝑘𝜙𝑘(𝑢)(𝑊(𝑢))
1

2|𝑝𝑑𝑢 = 0? 

M. Riesz [19]was the first to look into this kind of issue, focussing on the case of trigonometric 

functions. Later, Schauder [21], Kober [9], Caton and Hille [3] looked at other sets of functions. 

In this article the sequence of functions 𝜙𝑛(𝑢) are considered to be the orthogonal Hermite 

polynomials 𝐻𝑛(𝑢) with weight 𝑒−𝑢2
 satisfy  

 ∫
∞

−∞
𝐻𝑚(𝑢)𝐻𝑛(𝑢)𝑒−𝑢2

𝑑𝑢 = √𝜋2
𝑛

2𝑛! 𝛿𝑚𝑛. (2.3) 

The 𝑛𝑡ℎ  degree Hermite polynomials defined as 𝐻𝑛(𝑢) = (−1)𝑛𝑒𝑢2
(

𝑑

𝑑𝑢
)𝑛{𝑒−𝑢2

} [23]. The 

normalized Hermite functions of degree 𝑛 ∈ ℕ0 [4, 5, 6, 16] defined as,  

 𝜓𝑛(𝑢): = 𝑟𝑛𝐻𝑛(𝑢)𝑒−
𝑢2

2 , 𝑛 ≥ 0, 𝑢 ∈  ℝ, (2.4) 

where 𝑟𝑛 =
1

√2𝑛𝑛!√𝜋
 meet the orthonormal condition  

 ∫
∞

−∞
𝜓𝑚(𝑢)𝜓𝑛(𝑢)𝑑𝑢 = 𝛿𝑚𝑛. (2.5) 

These 𝜓𝑛(𝑢) form a basis in 𝐿𝑝(ℝ), 𝑝 ≥ 2 [24, 13]. 

 

Pollard [18] in 1948 showed that, if 𝑔(𝑢)𝑒
−𝑢2

2 ∈ 𝐿2(ℝ), then  

 ∫
∞

−∞
|𝑠𝑛(𝑢)|𝑝𝑒−𝑢2

𝑑𝑢 ≤ 𝐶 ∫
∞

−∞
|𝑔(𝑢)|2𝑒−𝑢2

𝑑𝑢, 

where, 𝑠𝑛 is the 𝑛𝑡ℎ partial sum of the Hermite polynomial series ∑∞
𝑘=0 𝑑𝑘𝐻𝑘(𝑢) for  

 𝑑𝑘: = 𝑟𝑘
2 ∫

∞

−∞
𝑔(𝑢)𝐻𝑘(𝑢)𝑒−𝑢𝑢

, 

𝑘 ∈ ℕ0. This suggests that  



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 ∥ 𝑠𝑛 − 𝑔 ∥2→ 0 (2.6) 

as 𝑛 → ∞ with ∥. ∥2 denoting the usual 𝐿2 norm on ℝ. 

This conclusion was extended to a larger class of functions 𝐿𝑝(ℝ), 
4

3
< 𝑝 < 4 by Askey and 

Wainger [1] in 1965. For measurable function 𝑔 such that 𝑔(𝑢)𝑒
−𝑢2

2 ∈ 𝐿𝑝(ℝ), 
4

3
< 𝑝 < 4, they 

proved the inequality  

 ∥ 𝑠𝑛(𝑢)𝑒−
𝑢2

2 ∥𝑝≤ 𝐶 ∥ 𝑔(𝑢)𝑒−
𝑢2

2 ∥𝑝, 

where 𝑠𝑛 = ∑𝑛
𝑘=0 𝑎𝑘𝜓𝑘(𝑢) and 𝑎𝑘 = ∫

∞

−∞
𝑓(𝑣)𝜓𝑘(𝑣)𝑑𝑣. This implies the mean Convergence  

 ∥ 𝑔(𝑢) − ∑𝑛
𝑘=0 𝑎𝑘𝜓𝑘(𝑢) ∥𝑝→ 0 (2.7) 

as 𝑛 → ∞ where ∥ 𝑔 ∥𝑝= {∫
∞

−∞
|𝑔|𝑝𝑑𝑢}

1

𝑝. 

In 1970, Muckenhoupt [14] generalized the Askey and Wainger [1] result for 𝑝 ∈ [1, ∞). He 

proved inequalities of the form  

 ∥ 𝑠𝑛(𝑢)𝑈(𝑢) ∥𝑝≤ 𝒞 ∥ 𝑔(𝑢)𝑊(𝑢) ∥𝑝, 

where 𝑈(𝑢), 𝑊(𝑢) are suitable weight functions. It lead to prove  

 ∥ (𝑠𝑛(𝑢) − 𝑔(𝑢))𝑈(𝑢) ∥𝑝→ 0, 

for every 𝑔 ∈ 𝐿𝑊(𝑢)
𝑝

 i.e., 𝑔(𝑢)𝑊(𝑢) ∈ 𝐿𝑝. If 𝑈(𝑢) = 𝑊(𝑢) = 𝑒
−𝑢2

2 , he obtained the result of Askey 

and Wainger [1] for 
4

3
< 𝑝 < 4. If 𝑈(𝑢, 𝑏) = 𝑒

−𝑢2

2 (1 + |𝑢|)𝑏 and 𝑊(𝑢, 𝐵) = 𝑒
−𝑢2

2 (1 + |𝑢|)𝐵 for 

different suitable numbers 𝑏 and 𝐵 such that 𝑏 < 𝐵, he obtained this result for 1 ≤ 𝑝 ≤
4

3
 and 𝑝 ≥

4. 𝑈(𝑢, 𝑏) and 𝑊(𝑢, 𝐵) are dense in 𝐿𝑝(ℝ)[14]. 

The random series considered in this article is expressed as  

 ∑∞
𝑘=0 𝑑𝑘ℛ𝑘(𝜔)𝐻𝑘(𝑢) (2.8) 

where 𝑑𝑘 represents scalars and ℛ𝑘 denotes random variables. 

The work of Nayak et al. [15] and Pattanayak and Sahoo[17] are followed to choose the 

random variables ℛ𝑘(𝜔), 𝑘 ∈ ℕ0 and to study the convergence of the random series (2.8). Suitable 

real numbers 𝑏 and 𝐵 are chosen such that 𝑏 < 𝐵, which implies,  

 ∥ (𝑠𝑛(𝑢) − 𝑔(𝑢))𝑈(𝑢, 𝑏) ∥2→ 0, 

by the result of Muckenhoupt (Theorem 6, [14]). In the first step, the existence of the stochastic 

integral  

 ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) 

is established for 𝑔 ∈ 𝐿𝑊(𝑢,𝐵)
2 (ℝ).  Since 𝑊(𝑢, 𝐵) is continuous for −

1

2
< 𝐵, 𝐻𝑘(𝑢)𝑊(𝑢, 𝐵) ∈

𝐿2(ℝ) and the integral ∫
∞

−∞
𝐻𝑘(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) exists. This integral is a random variable. 

Denote it as 𝒟𝑘(𝜔). Choose these 𝒟𝑘(𝜔) as the random coefficients in the series (2.8). The 

convergence of the series (2.8) in mean if the scalars  

 𝑑𝑘: = 𝑟𝑘
2 ∫

∞

−∞
𝑔(𝑢)𝑒−𝑢2

𝐻𝑘(𝑢)𝑑𝑢. 

are the FHC of a function 𝑔 in the weighted 𝐿𝑊(𝑢,𝐵)
2 (ℝ) space with weights 𝑊(𝑢, 𝐵) of the form 

𝑒
−𝑢2

2 (1 + |𝑢|)𝐵 for a suitable 𝐵. The stochastic integral  



Communications on Applied Nonlinear Analysis  

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 ∫
∞

−∞
𝑔(𝑢, 𝑣)𝑒

−𝑣2

2 (1 + |𝑣|)𝐵𝑑𝑋(𝑣, 𝜔), (2.9) 

is seen to be the sum function of this series. 

Throughout the sections 3 and 4 below, 𝑋(𝑢, 𝜔) is considered to be the SSP of index 𝜇 = 2 

and the weight functions 𝑈(𝑢, 𝑏) = 𝑒𝑥𝑝(−
1

2
𝑢2)(1 + |𝑢|)𝑏 , 𝑊(𝑢, 𝐵) = 𝑒𝑥𝑝(−

1

2
𝑢2)(1 + |𝑢|)𝐵 

where 𝑏 <
1

2
 and 𝐵 > −

1

2
 such that 𝑏 < 𝐵. 

 

3.  Existence of the stochastic integral  

The following result is required to prove the existence of the integral ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔). 

 

Lemma 1: [17] Suppose 𝑋(𝑢, 𝜔) is of index 𝜇, 𝜇 ∈ (1,2] and 𝑔 ∈ 𝐿𝑝[𝑎, 𝑏], 𝑝 ≥ 𝜇, 𝑠 ∈ ℝ then  

 𝐸(| ∫
𝑏

𝑎
𝑔(𝑢)𝑑𝑋(𝑢, 𝜔)|) ≤

4

𝜋(𝜇−1)
∫

𝑏

𝑎
|𝑔(𝑢)|𝜇𝑑𝑢 +

2

𝜋
∫

|𝑠|>1

1−𝑒𝑥𝑝(−|𝑠|𝜇 ∫
𝑏

𝑎
|𝑔(𝑢)|𝜇𝑑𝑢)

𝑠2 𝑑𝑠. 

 

Theorem 2:  If 𝑋(𝑢, 𝜔)  is of index 2 , and 𝑔(𝑢) ∈ 𝐿𝑊(𝑢,𝐵)
2 (ℝ) , then the integral 

∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) exists in mean .  

  

Proof: We are aware that 𝐶𝑐(ℝ) is dense in 𝐿2(ℝ). So for 𝑔 ∈ 𝐿𝑊(𝑢,𝐵)
2  there exist a sequence of 

functions {ℎ𝑘}  in 𝐶𝑐(ℝ)  such that (𝑔(𝑢)𝑊(𝑢, 𝐵) − ℎ𝑘) ∈ 𝐿2(ℝ)  and ∥ 𝑔𝑊(𝑢, 𝐵) − ℎ𝑘 ∥2 

approaches to 0 as 𝑘 → 0.  

Consider two functions ℎ𝑚 and ℎ𝑛 from this sequence {ℎ𝑘}.  

Without loss of generality assume that the compact support of ℎ𝑚 and ℎ𝑛 can be in [𝑎, 𝑏] and [c, d] 

respectively. So ℎ𝑚 and ℎ𝑛 can be considered to be in 𝐿2[𝑎, 𝑏] and 𝐿2[𝑐, 𝑑].  

Let [𝑝, 𝑞] be the smallest closed sub-interval of ℝ which contains [𝑎, 𝑏] ∪ [𝑐, 𝑑].  

Now both ℎ𝑚 and ℎ𝑛 can be considered to be in 𝐿2[𝑝, 𝑞]. Since ℎ𝑚 and ℎ𝑛 can be continuous, the 

stochastic integrals  

 ∫
𝑞

𝑝
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔) = ∫

𝑏

𝑎
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔) 

 = ∫
∞

−∞
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔) 

and  

 ∫
𝑞

𝑝
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) = ∫

𝑏

𝑎
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) 

 = ∫
∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) 

exists in the sense of mean[17].  

Denote  

 𝑌𝑚(𝜔): = ∫
𝑞

𝑝
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔)  

and 

 𝑌𝑛(𝜔): = ∫
𝑞

𝑝
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔). 

Now applying Lemma 1 for 𝜇 = 2, we get  

 𝐸|𝑌𝑛(𝜔) − 𝑌𝑚(𝜔)| 



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 = 𝐸(| ∫
𝑞

𝑝
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) − ∫

𝑞

𝑝
ℎ𝑚(𝑢)𝑑𝑋(𝑢, 𝜔)|) 

 = 𝐸(| ∫
𝑞

𝑝
(ℎ𝑛(𝑢) − ℎ𝑚(𝑡))𝑑𝑋(𝑢, 𝜔)|) 

 ≤
4

𝜋
∫

𝑞

𝑝
|ℎ𝑛(𝑢) − ℎ𝑚(𝑢)|2𝑑𝑢 +

2

𝜋
∫

|𝑠|>1

1−exp(−|𝑠|2 ∫
𝑞

𝑝 |ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢)

𝑠2 𝑑𝑠 

 ≤
4

𝜋
∫

∞

−∞
|ℎ𝑛(𝑢) − ℎ𝑚(𝑢)|2𝑑𝑢 +

2

𝜋
∫

|𝑠|>1

1−exp(−𝑐|𝑠|2 ∫
∞

−∞ |ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢)

𝑠2
𝑑𝑠. 

The integrand in the 2𝑛𝑑  integral is dominated by the integrable function 
1

𝑠2 over (−∞, −1] and 

[1, ∞) . Since ∥ ℎ𝑛(𝑢) − ℎ𝑚(𝑢) ∥2= ∫
∞

−∞
|ℎ𝑛(𝑢) − ℎ𝑚(𝑢)|2𝑑𝑢  approaches 0  as 𝑚, 𝑛 → ∞ , the 

2𝑛𝑑 integral converges to 0 by DCT and we obtained  

 lim
𝑚,𝑛→∞

𝐸|𝑌𝑛(𝜔) − 𝑌𝑚(𝜔)| = 0. 

𝑌𝑛(𝜔) is a Cauchy sequence in the sense of mean. Hence there exists a random variable 𝑌(𝜔) such 

that 𝐸|𝑌𝑛(𝜔) − 𝑌(𝜔)| = 0. This 𝑌(𝜔) is independent of the choice of the sequence of functions ℎ𝑛. 

In fact, if another sequence 𝑓𝑛 in 𝐶𝑐(ℝ) converges to 𝑔 i.e.  

 lim
𝑛→∞

∫
∞

−∞
|𝑓𝑛(𝑢) − 𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 = 0  𝑎𝑠  𝑛  →   ∞. 

Then  

 lim
𝑛→∞

∫
∞

−∞
|𝑓𝑛(𝑢) − ℎ𝑛(𝑢)|2𝑑𝑢 

 = lim
𝑛→∞

∫
∞

−∞
|𝑓𝑛(𝑢) − 𝑔(𝑢)𝑊(𝑢, 𝐵) + 𝑔(𝑢)𝑊(𝑢, 𝐵) − ℎ𝑛(𝑢)|2𝑑𝑢 

 = lim
𝑛→∞

∫
∞

−∞
|𝑓𝑛(𝑢) − 𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 + lim

𝑛→∞
∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵) − ℎ𝑛(𝑢)|2𝑑𝑢 

which converges to 0. Thus we obtain  

 lim
𝑛→∞

𝐸(| ∫
∞

−∞
𝑓𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) − 𝑌(𝜔)|) 

 = lim
𝑛→∞

𝐸(| ∫
∞

−∞
𝑓𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) − ∫

∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) + ∫

∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) −

𝑌(𝜔)|) 

 = lim
𝑛→∞

𝐸(| ∫
∞

−∞
𝑓𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) − ∫

∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔)|) +

lim
𝑛→∞

𝐸(| ∫
∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) − 𝑌(𝜔)|) 

 = 0   𝑏𝑦   𝐿𝑒𝑚𝑚𝑎   1. 

Hence the stochastic integral ∫
∞

−∞
ℎ𝑛(𝑢)𝑑𝑋(𝑢, 𝜔) converges uniquely to 𝑌(𝜔), in the sense of mean. 

Define this random variable 𝑌(𝜔) to be the stochastic integral, 𝑌(𝜔) = ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔).  

 This theorem implies the existence of the integral ∫
∞

−∞
𝐻𝑘(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) for 𝐵 >

−1

2
. 

The random variables 𝒟𝑘(𝜔) = ∫
∞

−∞
𝐻𝑘(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)  are found to be dependent. It is 

established by showing the fact that the characteristic function(CF) of (𝒟𝑘(𝜔) + 𝒟𝑙(𝜔)) is not equal 

to the product of CF of 𝒟𝑘(𝜔) and the CF 𝒟𝑙(𝜔). The CF of 𝒟𝑘(𝜔) is computed in the following 

theorem.  

 

 

 

 



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Theorem 3  The CF of ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) is 𝑒𝑥𝑝(−𝑐|𝑠|2 ∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢) for 

𝑔(𝑢) ∈ 𝐿𝑊(𝑢,𝐵)
2 (ℝ).  

 

Proof: As we know 𝐶𝑐(ℝ) is dense in 𝐿2(ℝ), there exist a sequence of functions {ℎ𝑘} in 𝐶𝑐(ℝ) for 

𝑔 ∈ 𝐿2(ℝ)  such that ∥ ℎ𝑘 − 𝑔𝑊(𝑢, 𝐵) ∥2→ 0 . Further it is known that the stochastic integrals 

∫
∞

−∞
ℎ𝑘𝑑𝑋(𝑢, 𝜔)  and ∫

∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)  exists by Theorem 2. Denote these random 

variables as 𝑌𝑘: = ∫
∞

−∞
ℎ𝑘(𝑢)𝑑𝑋(𝑢, 𝜔) and 𝑌: = ∫

∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) in mean.  

𝑌𝑘  converges to 𝑌  in mean ⇒  𝑌𝑘  converges to 𝑌  in law ⇒  distribution of 𝑌𝑘  weakly 

converges to distribution of 𝑌[12] .  

Now the CF of 𝑌𝑘: = exp(−𝑐|𝑠|2 ∫
∞

−∞
|ℎ𝑘(𝑢)|2𝑑𝑢).  

For 1 ≤ 𝑝 < ∞, it is true that ([20], page no. 75)  

 ∫
∞

−∞
||ℎ𝑘(𝑢)|2 − |𝑔(𝑢)|2|𝑑𝑢 ≤ 4𝑅 ∫

∞

−∞
|ℎ𝑘(𝑢) − 𝑔(𝑢)|2𝑑𝑢 → 0, 

and since 𝑊(𝑢, 𝐵) are dense in 𝐿2(ℝ), this implies  

 ∫
∞

−∞
|ℎ𝑘(𝑢)|2𝑑𝑢  𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠  𝑡𝑜  ∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 

 ⇒

exp(−𝑐|𝑠|2 ∫
∞

−∞
|ℎ𝑘(𝑢)|2𝑑𝑢)  𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠  𝑡𝑜  exp(−𝑐|𝑠|2 ∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢). 

LHS is the CF of 𝑌𝑘, which converges to the continuous function on the RSH.  

By continuity theorem([12], Theorem 1.3.7, page no. 15), RHS is the CF of the limiting 

function of 𝑌𝑘 , which is 𝑌 . This proves that the CF of ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)  is 

exp(−𝑐|𝑠|2 ∫
∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢) , implies the pointwise convergence of 𝐶𝑘(𝑠)  to 

exp(−𝑐|𝑠|2 ∫
∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢) as 𝑘 → ∞[2].  

 The following theorem proves that the random variables 𝒟𝑛(𝜔) are dependent.  

 

Theorem 4  The random variables 𝒟𝑛(𝜔) = ∫
∞

−∞
𝐻𝑛(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) are dependent.   

 

Proof: By Theorem 3, the CF of 𝒟𝑛(𝜔) is  

 exp(−𝑐|𝑠|2 ∫
∞

−∞
|𝐻𝑛(𝑥)𝑊(𝑢, 𝐵)|2𝑑𝑢). 

Hence, the CF of (𝒟𝑛(𝜔) + 𝒟𝑚(𝜔)) is  

 exp(−𝑐|𝑠|2 ∫
∞

−∞
|𝐻𝑛(𝑢)𝑊(𝑢, 𝐵) + 𝐻𝑚(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢), 

whereas the product of CF of 𝒟𝑛(𝜔) and the CF of 𝒟𝑚(𝜔) is  

 exp(−𝑐|𝑠|2 ∫
∞

−∞
|𝐻𝑛(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢)exp(−𝑐|𝑠|2 ∫

∞

−∞
|𝐻𝑚(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢) 

 = exp(−𝑐|𝑠|2 ∫
∞

−∞
(|𝐻𝑛(𝑢)𝑊(𝑢, 𝐵)|2 + |𝐻𝑚(𝑥)𝑊(𝑢, 𝐵)|2)𝑑𝑢). 

Since CF of (𝒟𝑛(𝜔) + 𝒟𝑚(𝜔)) is not equal to the product of CF of 𝒟𝑛(𝜔) and CF of 𝒟𝑚(𝜔), 

𝒟𝑛(𝜔) are dependent random variables.  

 

 

 



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4.  Convergence of random Fourier - Hermite series ∑∞
𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢) 

To prove the convergence of RFHS, we employ the following inequality. 

 

Lemma 5  Let 𝑔 be any function in 𝐿𝑊(𝑢,𝐵)
2 (ℝ) then  

 𝐸(| ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)|) 

 ≤
4

𝜋
∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 +

2

𝜋
∫

|𝑠|>1

1−exp(−|𝑠|2 ∫
∞

−∞ |𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢)

𝑠2 𝑑𝑠. 

  

 Its proof requires the following two results.  

 

Lemma 6 [22] A stable random variable 𝑋(𝑢, 𝜔) always satisfies the inequality 𝐸|𝑋|𝑖 < ∞ for all 

𝑖 ∈ (0, 𝜇), 0 < 𝜇 ≤ 2.   

 

Lemma 7 [7] If 𝛹 is the CF of a random variable 𝑋 and 𝐹(𝑋) is the distribution function of 𝑋 

then, 𝐸|𝑋| = ∫
∞

−∞
|𝑋|𝑑𝐹(𝑋) =

2

𝜋
∫

∞

−∞

1−𝑅𝑒𝛹(𝑠)

𝑠2 𝑑𝑠.   

 

Proof of Lemma 5: We know that, by Theorem 2, ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔) exists in mean.  

Now using Lemma 6 and 7, we have  

 𝐸(| ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)|) =

2

𝜋
∫

∞

−∞

1−𝑅𝑒Ψ(𝑠)

𝑠2 𝑑𝑠 

 =
2

𝜋
∫

|𝑠|≤1

1−𝑅𝑒Ψ(𝑠)

𝑠2
𝑑𝑠 +

2

𝜋
∫

|𝑠|>1

1−𝑅𝑒Ψ(𝑠)

𝑠2
𝑑𝑠. 

Here  

 ∫
|𝑠|≤1

1−𝑅𝑒Ψ(𝑠)

𝑠2 𝑑𝑠 = ∫
1

−1

1−exp(−|𝑠|2 ∫
∞

−∞ |𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢)

𝑠2 𝑑𝑠 

 ≤ ∫
1

−1

|𝑠|2 ∫
∞

−∞ |𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢

𝑠2 𝑑𝑠  (  ∵   1  −   𝑒−𝑢   <   𝑢  𝑓𝑜𝑟  𝑢  >   0) 

 = 2 ∫
1

0
𝑑𝑠 ∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 

 = 2 ∫
∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 

 Hence we have  

 𝐸(| ∫
∞

−∞
𝑔(𝑢)𝑊(𝑢, 𝐵)𝑑𝑋(𝑢, 𝜔)|) 

 ≤
4

𝜋
∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 +

2

𝜋
∫

|𝑠|>1

1−exp(−|𝑠|2 ∫
∞

−∞ |𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢)

𝑠2 𝑑𝑠. 

  

The following theorem establishes the convergence of the series ∑ 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢), to the 

integral  

 ∫
∞

−∞
𝑔(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔), (4.1) 

in the sense of mean, if  

 𝑑𝑘: = 𝑟𝑘
2 ∫

∞

−∞
𝑔(𝑣)𝐻𝑘(𝑣)𝑒−𝑣2

𝑑𝑣 (4.2) 

are the FHC of 𝑔 ∈ 𝐿𝑊(𝑣,𝐵)
2 (ℝ). Here 𝒟𝑘(𝜔) are defined as,  

 𝒟𝑘(𝜔): = ∫
∞

−∞
𝐻𝑘(𝑣)𝑊(𝑣, 𝐵)𝑑𝑋(𝑣, 𝜔). (4.3) 



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Its proof requires the following lemma, which is the statement of Theorem 1 and Theorem 6 of 

Muckenhoupt [14] for 𝑝 = 2. 

 

Lemma 8 [14] Let 𝑔 ∈ 𝐿𝑊(𝑣,   𝐵)
2 (ℝ) then,  

 ∫
∞

−∞
|𝑠𝑛(𝑔, 𝑢)𝑈(𝑢, 𝑏)|2𝑑𝑢 ≤ 𝒞 ∫

∞

−∞
|𝑔(𝑢)𝑊(𝑢, 𝐵)|2𝑑𝑢 

 and  

 ∥ (𝑠𝑛(𝑢) − 𝑔(𝑢))𝑈(𝑢, 𝑏) ∥2→ 0. (4.4) 

 

Theorem 9  For all measurable functions 𝑔 ∈ 𝐿𝑊(𝑢,𝐵)
2 (ℝ) , the series ∑∞

𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢) 

converges in mean to the integral (4.1).   

 

Proof: For 𝑔 ∈ 𝐿𝑊(𝑢,𝐵)
2 (ℝ), let the Fourier - Hermite series expansion of 𝑔 be ∑∞

𝑘=−∞ 𝑑𝑘𝐻𝑘(𝑢) [1]. 

Let its partial sum be 𝑠𝑛(𝑢): = ∑𝑛
𝑘=0 𝑑𝑘𝐻𝑘(𝑢) . Let 𝒮𝑛(𝑢, 𝜔) = ∑𝑛

𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢) be the 𝑛𝑡ℎ 

partial sum of the RFHS ∑∞
𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢).  

Now,  

 𝒮𝑛(𝑢, 𝜔) = ∑𝑛
𝑘=0 𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢) 

 = ∑𝑛
𝑘=0 𝑑𝑘(∫

∞

−∞
𝐻𝑘(𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔))𝐻𝑘(𝑢) 

 = ∫
∞

−∞
(∑𝑛

𝑘=0 𝑑𝑘𝐻𝑘(𝑣)𝐻𝑘(𝑢))𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔) 

Since, the series ∑𝑛
𝑘=0 𝑑𝑘𝐻𝑘(𝑥)𝐻𝑘(𝑣)  exists, let 𝑠𝑛(𝑢, 𝑣)  be the 𝑛𝑡ℎ  partial sum of the series 

∑∞
𝑘=0 𝑑𝑘𝐻𝑘(𝑢)𝐻𝑘(𝑣). This implies,  

 𝒮𝑛(𝑢, 𝜔) = ∫
∞

−∞
𝑠𝑛(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔). 

By Lemma 2, we know that, ∫
∞

−∞
𝑔(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔) exists in mean.  

Now,  

 𝐸(|𝑆𝑛(𝑢, 𝜔) − ∫
∞

−∞
𝑔(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔)|) 

 = 𝐸(| ∫
∞

−∞
𝑠𝑛(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔) − ∫

∞

−∞
𝑔(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔)|) 

 = 𝐸(| ∫
∞

−∞
(𝑠𝑛(𝑢, 𝑣)𝑈(𝑣, 𝑏) − 𝑔(𝑢, 𝑣)𝑈(𝑣, 𝑏)𝑑𝑋(𝑣, 𝜔)|) 

 ≤
4

𝜋
∫

∞

−∞
|(𝑠𝑛(𝑢, 𝑣) − 𝑔(𝑢, 𝑣))𝑈(𝑣, 𝑏)|2𝑑𝑣 

 +
2

𝜋
∫

|𝑠|>1

1 − exp(−|𝑠|2 ∫
∞

−∞
|(𝑠𝑛(𝑢,   𝑣)−𝑔(𝑢,   𝑣))𝑈(𝑣,   𝑏)|2𝑑𝑣)

𝑠2 𝑑𝑠 

Lemma 8 and the dominance of 
1

𝑠2 lead both of these integrals to tend to zero. Thus, the theorem is 

established.  

  

Acknowledgments 

This research work was supported by UGC (RGNF) with letter 

no-F./2015-16/RGNF-SC-2015-16-SC-ORI-20053. 

 

References 

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Communications on Applied Nonlinear Analysis  

ISSN: 1074-133X 

Vol 31 No. 2s (2024) 

 

 

704 https://internationalpubls.com 

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