Convergence In Distribution Of Components Implies Convergence In Distribution Of Vector?

Introduction

In probability theory, the concept of convergence in distribution is a crucial aspect of understanding the behavior of random variables. When we say that a sequence of random variables $X_n$ converges in distribution to a random variable $X$, denoted as $X_n \xrightarrow{d} X$, it means that the cumulative distribution functions (CDFs) of $X_n$ converge to the CDF of $X$ at all continuity points. In this article, we will explore the relationship between the convergence in distribution of components and the convergence in distribution of a vector.

The Problem Statement

Let $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} y$, where $y$ is some constant. Suppose for generality that $X \in \mathbb{R}^n$, and $y \in \mathbb{R}^m$. Our goal is to determine whether the convergence in distribution of the components $X_n$ and $Y_n$ implies the convergence in distribution of the vector $(X_n, Y_n)$ to the vector $(X, y)$.

Theoretical Background

To tackle this problem, we need to understand the concept of convergence in distribution and its relationship with the convergence of the CDFs. Let's start by defining the CDF of a random variable $X$ as $F_X(x) = P(X \leq x)$. The convergence in distribution of $X_n$ to $X$ is denoted as $X_n \xrightarrow{d} X$ and is equivalent to the statement that $F_{X_n}(x) \to F_X(x)$ for all continuity points $x$ of $F_X$.

The Convergence in Distribution of Components

We are given that $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} y$. This means that the CDFs of $X_n$ and $Y_n$ converge to the CDFs of $X$ and $y$, respectively, at all continuity points. We can write this as:

$$F_{X_n}(x) \to F_X(x) \quad \text{and} \quad F_{Y_n}(y) \to F_y(y)$$

for all continuity points $x$ and $y$.

The Convergence in Distribution of the Vector

Now, let's consider the vector $(X_n, Y_n)$. We want to determine whether the convergence in distribution of the components $X_n$ and $Y_n$ implies the convergence in distribution of the vector $(X_n, Y_n)$ to the vector $(X, y)$. To do this, we need to examine the CDF of the vector $(X_n, Y_n)$.

The CDF of the vector $(X_n, Y_n)$ is given by:

$$F_{(X_n, Y_n)}(x, y) = P(X_n \leq x, Y_n \leq y)$$

Using the definition of the CDF, we can rewrite this as:

$$F_{(X_n, Y_n)}(x, y) = P(X_n \leq x)P(Y_n \leq y)$$

Now, we can use the fact that $_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} y$ to conclude that:

$$F_{(X_n, Y_n)}(x, y) \to F_X(x)F_y(y)$$

for all continuity points $x$ and $y$.

Conclusion

In conclusion, we have shown that the convergence in distribution of the components $X_n$ and $Y_n$ implies the convergence in distribution of the vector $(X_n, Y_n)$ to the vector $(X, y)$. This result is a direct consequence of the definition of convergence in distribution and the properties of the CDFs.

Implications

This result has important implications in probability theory and statistics. For example, it can be used to establish the convergence in distribution of random vectors in higher dimensions. Additionally, it can be used to study the asymptotic behavior of random variables and vectors in various applications, such as finance, engineering, and social sciences.

Future Directions

There are several directions for future research. One possible direction is to extend this result to more general types of convergence, such as convergence in probability or convergence in mean. Another direction is to study the convergence in distribution of random vectors with more complex structures, such as random matrices or random tensors.

References

• Billingsley, P. (1995). Probability and Measure (3rd ed.). Wiley.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Shao, J. (2003). Mathematical Statistics. Springer.

Appendix

The appendix contains the proof of the main result. The proof is based on the definition of convergence in distribution and the properties of the CDFs.

Proof of the Main Result

Let $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} y$. We want to show that $(X_n, Y_n) \xrightarrow{d} (X, y)$. To do this, we need to show that $F_{(X_n, Y_n)}(x, y) \to F_{(X, y)}(x, y)$ for all continuity points $x$ and $y$.

Using the definition of the CDF, we can write:

$$F_{(X_n, Y_n)}(x, y) = P(X_n \leq x, Y_n \leq y)$$

$$= P(X_n \leq x)P(Y_n \leq y)$$

$$\to F_X(x)F_y(y)$$

for all continuity points $x$ and $y$.

Introduction

In our previous article, we explored the relationship between the convergence in distribution of components and the convergence in distribution of a vector. We showed that if $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} y$, then $(X_n, Y_n) \xrightarrow{d} (X, y)$. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is convergence in distribution?

A: Convergence in distribution is a concept in probability theory that describes the behavior of random variables. It states that a sequence of random variables $X_n$ converges in distribution to a random variable $X$ if the cumulative distribution functions (CDFs) of $X_n$ converge to the CDF of $X$ at all continuity points.

Q: What is the difference between convergence in distribution and convergence in probability?

A: Convergence in distribution and convergence in probability are two different concepts in probability theory. Convergence in distribution describes the behavior of the CDFs of random variables, while convergence in probability describes the behavior of the probability measures of random variables.

Q: Can you provide an example of convergence in distribution?

A: Yes, consider a sequence of random variables $X_n$ that are uniformly distributed on the interval $[0, 1/n]$. As $n$ increases, the CDF of $X_n$ converges to the CDF of a random variable $X$ that is uniformly distributed on the interval $[0, 1]$. This is an example of convergence in distribution.

Q: How does the convergence in distribution of components imply the convergence in distribution of a vector?

A: The convergence in distribution of components implies the convergence in distribution of a vector because the CDF of the vector is the product of the CDFs of the components. If the CDFs of the components converge to the CDFs of the limiting random variables, then the CDF of the vector converges to the CDF of the limiting vector.

Q: Can you provide a counterexample to show that the converse is not true?

A: Yes, consider a sequence of random vectors $(X_n, Y_n)$ that are uniformly distributed on the square $[0, 1] \times [0, 1/n]$. As $n$ increases, the CDF of the vector $(X_n, Y_n)$ converges to the CDF of a random vector $(X, Y)$ that is uniformly distributed on the square $[0, 1] \times [0, 1]$. However, the CDFs of the components $X_n$ and $Y_n$ do not converge to the CDFs of the limiting random variables $X$ and $Y$. This is a counterexample to show that the converse is not true.

Q: What are some applications of convergence in distribution?

A: Convergence in distribution has many applications in probability theory and statistics. Some examples include:

• Establishing the convergence of random variables in higher dimensions Studying the asymptotic behavior of random variables and vectors in various applications, such as finance, engineering, and social sciences
• Developing statistical inference methods, such as hypothesis testing and confidence intervals

Q: Can you provide some references for further reading?

A: Yes, some references for further reading include:

• Billingsley, P. (1995). Probability and Measure (3rd ed.). Wiley.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Shao, J. (2003). Mathematical Statistics. Springer.

Conclusion

In conclusion, we have answered some frequently asked questions related to the convergence in distribution of components and the convergence in distribution of a vector. We hope that this article has provided a helpful resource for readers who are interested in this topic.