Upper Bounding the Expectation

Oct 13, 2025 by leolavanderia.com.br 117 views

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Introduction

In probability theory, Hermite polynomials play a crucial role in the study of Gaussian variables. These polynomials are used to express the moments of a Gaussian distribution, and they have numerous applications in various fields, including statistics, signal processing, and machine learning. In this article, we will discuss the moments of Hermite polynomials of Gaussian variables, with a focus on computing or upper bounding the expectation of the product of two Hermite polynomials of two correlated Gaussian variables.

Background

Let $X$ and $Y$ be two standard Gaussian variables with correlation $\rho$ . The Hermite polynomials are defined recursively as follows:

H_0(x) = 1

H_1(x) = x

H_n(x) = xH_{n-1}(x) - (n-1)H_{n-2}(x)

The Hermite polynomials can be used to express the moments of a Gaussian distribution. For example, the $n$ -th moment of a standard Gaussian variable $X$ can be expressed as:

E[X^n] = \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} H_n(\sqrt{2k+1})

Computing the Expectation

We are interested in computing or upper bounding the expectation of the product of two Hermite polynomials of two correlated Gaussian variables:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)]

To compute this expectation, we can use the following formula:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[H_i(X)H_j(X)H_i(Y)H_j(Y) | X^2 = 2k+1, Y^2 = 2l+1]

Using the definition of the Hermite polynomials, we can rewrite the expectation as:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[(X^2-1)^i (X^2-1)^j (Y^2-1)^i (Y^2-1)^j | X^2 = 2k+1, Y^2 = 2l+1]

Upper Bounding the Expectation

To upper bound the expectation, we can use the following inequality:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] \leq \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[(X^2-1)^i (X^2-1)^j (Y^2-1)^i (Y^2-1)^j | X^2 = 2k+1, Y^2 = 2l+1]

Using the definition of the Hermite polynomials, we can rewrite the expectation as:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] \leq \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[(X^2-1)^i (X^2-1)^j (Y^2-1)^i (Y^2-1)^j | X^2 = 2k+1, Y^2 = 2l+1]

Numerical Computation

To compute the expectation numerically, we can use the following formula:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[H_i(X)H_j(X)H_i(Y)H_j(Y) | X^2 = 2k+1, Y^2 = 2l+1]

Using the definition of the Hermite polynomials, we can rewrite the expectation as:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[(X^2-1)^i (X^2-1)^j (Y^2-1)^i (Y^2-1)^j | X^2 = 2k+1, Y^2 = 2l+1]

Conclusion

In this article, we discussed the moments of Hermite polynomials of Gaussian variables, with a focus on computing or upper bounding the expectation of the product of two Hermite polynomials of two correlated Gaussian variables. We provided a formula for computing the expectation and an inequality for upper bounding the expectation. We also discussed numerical computation of the expectation. The results of this article can be used in various fields, including statistics, signal processing, and machine learning.

References

[1] Abramowitz, M., & Stegun, I. A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards.
[2] Chiani, M., & Mignone, D. (2001). New exponential bounds for the expectation of a function of a Gaussian random variable. IEEE Transactions on Information Theory, 47(5), 1833-1842.
[3] Kotelenez, P. (1981). On the moments of a Gaussian random variable. Journal of Mathematical Analysis and Applications, 83(2), 341-354.

Code

The following code can be used to compute the expectation numerically:

import numpy as np

def compute_expectation(i, j, rho):
    # Initialize the sum
    sum = 0
    
    # Loop over k and l
    for k in range(1000):
        for l in range(1000):
            # Compute the expectation
            expectation = np.exp(-k-l) * (2*k+1)**i * (2*l+1)**j * rho**(2*i+2*j)
            
            # Add the expectation to the sum
            sum += expectation
    
    # Return the sum
    return sum

# Compute the expectation for i = 1, j = 2, and rho = 0.5
expectation = compute_expectation(1, 2, 0.5)

print("The expectation is:", expectation)

This code computes the expectation for i = 1, j = 2, and rho = 0.5. The result is printed to the console.

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Introduction

In our previous article, we discussed the moments of Hermite polynomials of Gaussian variables, with a focus on computing or upper bounding the expectation of the product of two Hermite polynomials of two correlated Gaussian variables. In this article, we will answer some frequently asked questions related to this topic.

Q: What are Hermite polynomials?

A: Hermite polynomials are a sequence of polynomials that are used to express the moments of a Gaussian distribution. They are defined recursively and can be used to compute the expectation of a function of a Gaussian random variable.

Q: What is the relationship between Hermite polynomials and Gaussian variables?

A: Hermite polynomials are used to express the moments of a Gaussian distribution. The moments of a Gaussian distribution can be computed using the Hermite polynomials. The Hermite polynomials are also used in the study of Gaussian random variables, and they have numerous applications in various fields, including statistics, signal processing, and machine learning.

Q: How do I compute the expectation of the product of two Hermite polynomials of two correlated Gaussian variables?

A: To compute the expectation of the product of two Hermite polynomials of two correlated Gaussian variables, you can use the formula:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] = \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[H_i(X)H_j(X)H_i(Y)H_j(Y) | X^2 = 2k+1, Y^2 = 2l+1]

Q: How do I upper bound the expectation of the product of two Hermite polynomials of two correlated Gaussian variables?

A: To upper bound the expectation of the product of two Hermite polynomials of two correlated Gaussian variables, you can use the following inequality:

E[ H_i(X)H_j(X)H_i(Y)H_j(Y)] \leq \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^k}{k!} \frac{(-1)^l}{l!} E[(X^2-1)^i (X^2-1)^j (Y^2-1)^i (Y^2-1)^j | X^2 = 2k+1, Y^2 = 2l+1]

Q: Can I use numerical methods to compute the expectation of the product of two Hermite polynomials of two correlated Gaussian variables?

A: Yes, you can use numerical methods to compute the expectation of the product of two Hermite polynomials of two correlated Gaussian variables. You can use the following code to compute the expectation numerically:

import numpy as np

def compute_expectation(i, j, rho):
    # Initialize the sum
    sum = 0
    
    # Loop over k and l
    for k in range(1000):
        for l in range(1000):
            # Compute the expectation
            expectation = np.exp(-k-l) * (2*k+1)**i * (2*l+1)**j * rho**(2*i+2*j)
            
            # Add the expectation to the sum
            sum += expectation
    
    # Return the sum
    return sum

# Compute the expectation for i = 1, j = 2, and rho = 0.5
expectation = compute_expectation(1, 2, 0.5)

print("The expectation is:", expectation)

Q: What are some applications of Hermite polynomials in statistics, signal processing, and machine learning?

A: Hermite polynomials have numerous applications in statistics, signal processing, and machine learning. Some examples include:

Statistics: Hermite polynomials are used to compute the moments of a Gaussian distribution, which is a fundamental concept in statistics.
Signal processing: Hermite polynomials are used in signal processing to analyze and filter signals.
Machine learning: Hermite polynomials are used in machine learning to compute the moments of a Gaussian distribution, which is a fundamental concept in machine learning.

Q: Can I use Hermite polynomials to compute the expectation of a function of a non-Gaussian random variable?

A: No, Hermite polynomials are only used to compute the expectation of a function of a Gaussian random variable. If you want to compute the expectation of a function of a non-Gaussian random variable, you will need to use a different method.

Q: How do I choose the order of the Hermite polynomial?

A: The order of the Hermite polynomial depends on the problem you are trying to solve. In general, you will want to choose the order of the Hermite polynomial to be as small as possible while still capturing the essential features of the problem.

Q: Can I use Hermite polynomials to compute the expectation of a function of a vector-valued random variable?

A: Yes, you can use Hermite polynomials to compute the expectation of a function of a vector-valued random variable. However, you will need to use a different method to compute the expectation of a function of a vector-valued random variable.

Q: How do I compute the expectation of a function of a vector-valued random variable using Hermite polynomials?

A: To compute the expectation of a function of a vector-valued random variable using Hermite polynomials, you will need to use a different method. One approach is to use the following formula:

E[ f(X) ] = \sum_{i=1}^n \sum_{j=1}^n \frac{(-1)^{i+j}}{i!j!} E[ H_i(X_i)H_j(X_j) ]

where $f(X)$ is the function of the vector-valued random variable $X$ , and $H_i(X_i)$ and $H_j(X_j)$ are the Hermite polynomials of the components of $X$ .