Show Convergence To Standard Normal Distribution (check Of Proof)

Introduction

In the realm of probability theory, the Central Limit Theorem (CLT) stands as a cornerstone, illuminating the behavior of sums of independent random variables. This article delves into the intricacies of demonstrating convergence to the standard normal distribution, specifically focusing on a scenario involving an independent and identically distributed (iid) sequence of non-degenerate random variables. We will explore the conditions under which such convergence occurs and meticulously examine the key steps involved in proving this fundamental concept. Our exploration will center around the problem statement: "Let $X_k$, $k \geq 1$ be an iid sequence of non-degenerate random variables with $X_1 \in \mathcal{L}^\infty$. Show that for each $\alpha \geq -\frac{1}{2}$..." This article aims to provide a comprehensive understanding of the proof techniques and the underlying principles governing the convergence to the standard normal distribution, making it an invaluable resource for students, researchers, and practitioners alike. The journey through this proof will not only solidify your understanding of the CLT but also enhance your ability to tackle related problems in probability and statistics. By breaking down the complex concepts into manageable segments, we aim to demystify the process and empower you with the knowledge to confidently navigate the world of probability distributions.

Problem Statement and Initial Considerations

To rigorously address the problem of showing convergence to the standard normal distribution, let's first dissect the problem statement: "Let $X_k$, $k \geq 1$ be an iid sequence of non-degenerate random variables with $X_1 \in \mathcal{L}^\infty$. Show that for each $\alpha \geq -\frac{1}{2}$...". The essence of this statement lies in the properties of the sequence {$X_k$}. The term "iid" signifies that the random variables are independent and identically distributed. This is a critical assumption, as the CLT heavily relies on the independence of the variables and their shared statistical characteristics. "Non-degenerate" implies that the variables are not constant; they possess variability and are not confined to a single value. The condition $X_1 \in \mathcal{L}^\infty$ introduces the concept of boundedness. It means that $X_1$ belongs to the space of essentially bounded random variables, indicating that it is bounded almost surely. This condition plays a vital role in controlling the tails of the distribution and ensuring that certain moments exist.

Understanding these initial conditions is paramount. The independence allows us to leverage powerful tools like characteristic functions, which transform sums into products. The identical distribution ensures that the variables share the same mean and variance, which are crucial parameters for standardization. The boundedness condition guarantees that higher-order moments are finite, simplifying the analysis of the characteristic function's behavior. As we embark on the proof, these conditions will serve as guideposts, illuminating the path toward demonstrating the convergence to the standard normal distribution. The next steps will involve standardizing the variables, analyzing the characteristic function, and applying appropriate limit theorems to establish the desired convergence. Each step is carefully constructed to build upon the previous one, leading us to a robust and comprehensive understanding of the theorem's application.

Standardization and Characteristic Functions

Before diving deep into the proof, standardization is paramount. We are dealing with random variables {$X_k$}, each having a mean and a standard deviation. Standardization involves transforming these variables into a new set with a mean of 0 and a standard deviation of 1. This process simplifies the analysis and allows us to compare the distribution of the sum with the standard normal distribution, which inherently has these properties. The standardization is achieved by subtracting the mean and dividing by the standard deviation. If $\mu$ is the mean of $X_1$ and $\sigma$ is its standard deviation, the standardized variable $Z_k$ can be expressed as $Z_k = (X_k - \mu) / \sigma$. By standardizing, we effectively remove the location and scale parameters, allowing us to focus solely on the shape of the distribution. This step is crucial because the Central Limit Theorem is fundamentally about the shape of the distribution of the sum of random variables, not its location or scale.

The characteristic function is another critical tool in this proof. It is essentially the Fourier transform of the probability density function of a random variable. For a random variable X, the characteristic function $\phi_X(t)$ is defined as $\phi_X(t) = E[e^{itX}]$, where E denotes the expected value and i is the imaginary unit. Characteristic functions are particularly useful because they transform sums of independent random variables into products. If we have independent random variables $Z_1, Z_2, ..., Z_n$, the characteristic function of their sum is the product of their individual characteristic functions. This property significantly simplifies the analysis of sums of random variables. Furthermore, the characteristic function uniquely determines the distribution of a random variable. If we can show that the characteristic function of a standardized sum converges to the characteristic function of the standard normal distribution, we can conclude that the distribution of the standardized sum converges to the standard normal distribution. The characteristic function of the standard normal distribution is $e^{-t^2/2}$. Thus, our goal is to show that the characteristic function of the standardized sum of the $X_k$ converges to this expression as n approaches infinity. This step-by-step process, from standardization to leveraging characteristic functions, is a cornerstone of demonstrating convergence in probability theory, particularly in the context of the Central Limit Theorem.

Convergence of the Characteristic Function

The heart of the proof lies in demonstrating the convergence of the characteristic function of the standardized sum to the characteristic function of the standard normal distribution. Let's denote the sum of the first n standardized variables as $S_n = \frac{1}{\sqrt{n}} \sum_{k=1}^{n} Z_k$, where $Z_k = (X_k - \mu) / \sigma$ are the standardized random variables. The characteristic function of $S_n$ is given by $\phi_{S_n}(t) = E[e^{itS_n}]$. Since the $Z_k$ are independent, we can express $\phi_{S_n}(t)$ as a product of individual characteristic functions:

$$\phi_{S_n}(t) = E\left[e^{it(\frac{1}{\sqrt{n}} \sum_{k=1}^{n} Z_k)}\right] = \prod_{k=1}^{n} E\left[e^{itZ_k/\sqrt{n}}\right] = \left(\phi_Z(\frac{t}{\sqrt{n}})\right)^n$$

where $\phi_Z(t)$ is the characteristic function of a single standardized variable Z. Our goal is to show that as n approaches infinity, $\phi_{S_n}(t)$ converges to $e^{-t^2/2}$, which is the characteristic function of the standard normal distribution. To achieve this, we can analyze the behavior of $\phi_Z(\frac{t}{\sqrt{n}})$ for large n. We can use a Taylor series expansion to approximate $\phi_Z(x)$ around 0. Since $E[Z] = 0$ and $Var(Z) = 1$, the Taylor expansion of $\phi_Z(t)$ is given by:

$$\phi_Z(t) = 1 + itE[Z] - \frac{t^2}{2}E[Z^2] + o(t^2) = 1 - \frac{t^2}{2} + o(t^2)$$

where $o(t^2)$ represents terms that go to zero faster than $t^2$ as t approaches 0. Substituting $\frac{t}{\sqrt{n}}$ for t, we get:

$$\phi_Z(\frac{t}{\sqrt{n}}) = 1 - \frac{t^2}{2n} + o(\frac{t^2}{n})$$

Now, we raise this expression to the power of n to obtain the characteristic function of $S_n$:

$$\phi_{S_n}(t) = \left(1 - \frac{t^2}{2n} + o(\frac{t^2}{n})\right)^n$$

Taking the limit as n approaches infinity, we can use the result that $\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x$ to obtain:

$$\lim_{n \to \infty} \phi_{S_n}(t) = \lim_{n \to \infty} \left(1 - \frac{t^2}{2n} + o(\frac{t^2}{n})\right)^n = e^{-t^2/2}$$

This demonstrates that the characteristic function of the standardized sum $S_n$ converges to the characteristic function of the standard normal distribution. This convergence is a crucial step in proving the Central Limit Theorem, as it implies that the distribution of the standardized sum converges weakly to the standard normal distribution.

Application of the Continuity Theorem

Having established the convergence of the characteristic function, the next crucial step is to invoke the Continuity Theorem for characteristic functions. This theorem provides the bridge between the convergence of characteristic functions and the convergence in distribution of the corresponding random variables. The Continuity Theorem states that if a sequence of characteristic functions {$\phi_{X_n}(t)$} converges pointwise to a function $\phi(t)$ that is continuous at t = 0, then $\phi(t)$ is the characteristic function of some random variable X, and the sequence of random variables {$X_n$} converges in distribution to X. In our case, we have shown that the characteristic function of the standardized sum, $\phi_{S_n}(t)$, converges pointwise to $e^{-t^2/2}$, which is the characteristic function of the standard normal distribution. Furthermore, $e^{-t^2/2}$ is continuous for all t, including t = 0. Therefore, we can apply the Continuity Theorem.

By applying the Continuity Theorem, we conclude that the standardized sum $S_n$ converges in distribution to the standard normal distribution. This means that for any bounded continuous function g, we have:

$$\lim_{n \to \infty} E[g(S_n)] = E[g(Z)]$$

where Z is a standard normal random variable. This result is the essence of the Central Limit Theorem: the sum of a large number of independent and identically distributed random variables, when properly standardized, converges in distribution to the standard normal distribution, a cornerstone of statistical inference. The Continuity Theorem acts as the final link in this chain of reasoning, allowing us to translate the convergence of characteristic functions into the convergence of distributions. This step underscores the power and elegance of using characteristic functions as a tool for analyzing the behavior of sums of random variables.

Addressing the Condition $\alpha \geq -\frac{1}{2}$

The original problem statement included the condition "for each $\alpha \geq -\frac{1}{2}$...". While the preceding discussion has focused on the core proof of convergence to the standard normal distribution via the Central Limit Theorem, this condition likely pertains to a specific aspect of the problem that has been omitted from the initial query. To fully address this condition, we need more context regarding the complete problem statement. However, we can speculate on its possible relevance based on common scenarios in probability theory.

One possibility is that $\alpha$ relates to a moment condition or a rate of convergence. For instance, it might be part of a statement about the convergence of moments. If the problem involves showing that $E[|S_n|^\alpha]$ converges to $E[|Z|^\alpha]$ for $\alpha \geq -\frac{1}{2}$, where $S_n$ is the standardized sum and Z is a standard normal random variable, then the condition ensures that we are considering moments that exist. Negative moments can be more delicate to handle, and the restriction $\alpha \geq -\frac{1}{2}$ might be necessary to ensure the finiteness of the moments. Another possibility is that $\alpha$ is related to a Berry-Esseen type result, which provides a bound on the rate of convergence to the normal distribution. Such bounds often involve conditions on moments, and $\alpha$ might appear in the expression for the bound or in the conditions required for the bound to hold. Without the full problem statement, it's challenging to provide a definitive explanation. However, the most likely interpretation is that the condition $\alpha \geq -\frac{1}{2}$ is a technical requirement related to the existence of moments or the validity of a rate of convergence result. To fully address this aspect, the complete problem statement is essential. This highlights the importance of providing the entire context when seeking assistance with mathematical problems, as even seemingly minor conditions can play a crucial role in the solution.

Conclusion

In conclusion, demonstrating convergence to the standard normal distribution through the Central Limit Theorem is a multifaceted process that involves several key steps. We began by understanding the problem statement, emphasizing the importance of independent and identically distributed random variables and the condition of boundedness. The standardization of variables was crucial for aligning the scales and focusing on the shape of the distribution. The use of characteristic functions allowed us to transform the sum of random variables into a product, simplifying the analysis. Through Taylor series expansion and limit arguments, we showed that the characteristic function of the standardized sum converges to the characteristic function of the standard normal distribution. The application of the Continuity Theorem then bridged the gap between the convergence of characteristic functions and the convergence in distribution.

The condition $\alpha \geq -\frac{1}{2}$, while not fully addressed due to the lack of the complete problem statement, likely pertains to technical requirements related to moment conditions or rates of convergence. The detailed exploration of these steps provides a robust understanding of the CLT and its application in demonstrating convergence. The Central Limit Theorem stands as a powerful tool in probability theory and statistics, with far-reaching implications in various fields. This article has aimed to demystify the proof process, making it accessible and understandable. The journey from the initial problem statement to the final conclusion underscores the elegance and power of mathematical reasoning in unraveling complex phenomena. By mastering the concepts and techniques presented here, students, researchers, and practitioners can confidently apply the Central Limit Theorem in their work and further explore the rich landscape of probability theory.