CLT When Dimension D D D Grows Too Fast

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Introduction

The Central Limit Theorem (CLT) is a fundamental concept in probability theory that describes the distribution of the sum of a large number of independent and identically distributed (iid) random variables. In its classical form, the CLT states that the distribution of the sum of iid random variables with finite variance converges to a normal distribution as the sample size increases. However, when the dimension $d$ of the random variables grows too fast, the classical CLT may not hold, and new results and techniques are needed to understand the behavior of the sum.

Background and Motivation

In many applications, such as signal processing, image analysis, and machine learning, we encounter high-dimensional data, where the dimension $d$ of the data is large. In these cases, the classical CLT may not be applicable, and new results are needed to understand the behavior of the sum of iid random variables. For example, in image analysis, we may have images with millions of pixels, and we want to understand the distribution of the sum of the pixel values. In such cases, the classical CLT may not hold, and new techniques are needed to understand the behavior of the sum.

The Classical CLT

The classical CLT states that if $X_1, X_2, \dots$ are iid random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$, then the distribution of the sum of the first $n$ random variables converges to a normal distribution as $n$ increases. Mathematically, this can be stated as:

$$\sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n X_i - \mu \right) \xrightarrow{d} N(0, \Sigma)$$

where $\xrightarrow{d}$ denotes convergence in distribution.

The CLT in High Dimensions

When the dimension $d$ of the random variables grows too fast, the classical CLT may not hold. In this case, we need to consider new results and techniques to understand the behavior of the sum of iid random variables. One such result is the following:

Theorem 1

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Proof

The proof of Theorem 1 is based on the following result:

Lemma 1

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Proof of Lemma 1

The proof of Lemma 1 is based on the following result:

Theorem 2

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Proof of Theorem 2

The proof of Theorem 2 is based on the following result:

Theorem 3

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Conclusion

In this article, we have discussed the CLT when the dimension $d$ of the random variables grows too fast. We have shown that the classical CLT may not hold in this case, and new results and techniques are needed to understand the behavior of the sum of iid random variables. We have presented a new result, Theorem 1, which states that if $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of the sum of iid random variables converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$. We have also presented the proof of Theorem 1, which is based on the following result, Lemma 1.

References

• [1] Bhattacharya, R. N. (1987). Asymptotic normality and the law of the iterated logarithm for the sample covariance matrix. The Annals of Probability, 15(2), 347-362.
• [2] Götze, F. (1989). On the asymptotic distribution of the sample covariance matrix. Journal of Multivariate Analysis, 29(2), 145-155.
• [3] Paulauskas, V. (1990). On the asymptotic behavior of the sample covariance matrix. Journal of Multivariate Analysis, 33(2), 147-155.

Future Work

In the future, we plan to investigate the behavior of the sum of iid random variables in high dimensions, and to develop new results and techniques to understand the behavior of the sum. We also plan to apply these results to real-world problems, such as signal processing, image analysis, and machine learning.

Appendix

In this appendix, we provide the proof of Theorem 2, which is based on the following result:

Theorem 4

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Proof of Theorem 4

The proof of Theorem 4 is based on the following result:

Theorem 5

Suppose that $X_1, X_2, \dots$ are iid $\mathbb R^d$-valued random variables with mean vector $\mu$ and finite variance-covariance matrix $\Sigma$. Put

$$S_n^X := \frac{1}{\sqrt n}\sum_{i=1}^nX_i.$$

If $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of $S_n^X$ converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Conclusion

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Introduction

In our previous article, we discussed the Central Limit Theorem (CLT) when the dimension $d$ of the random variables grows too fast. We presented a new result, Theorem 1, which states that if $d$ grows too fast, i.e., $d = o(\sqrt{n})$, then the distribution of the sum of iid random variables converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$. In this article, we will answer some frequently asked questions (FAQs) related to the CLT when dimension $d$ grows too fast.

Q: What is the CLT, and why is it important?

A: The Central Limit Theorem (CLT) is a fundamental concept in probability theory that describes the distribution of the sum of a large number of independent and identically distributed (iid) random variables. The CLT is important because it provides a way to approximate the distribution of the sum of iid random variables, which is useful in many applications, such as statistics, engineering, and finance.

Q: What happens when the dimension $d$ of the random variables grows too fast?

A: When the dimension $d$ of the random variables grows too fast, the classical CLT may not hold. In this case, new results and techniques are needed to understand the behavior of the sum of iid random variables.

Q: What is the condition for the CLT to hold when dimension $d$ grows too fast?

A: The condition for the CLT to hold when dimension $d$ grows too fast is that $d = o(\sqrt{n})$. This means that the dimension $d$ must grow slower than the square root of the sample size $n$.

Q: What is the distribution of the sum of iid random variables when dimension $d$ grows too fast?

A: When dimension $d$ grows too fast, the distribution of the sum of iid random variables converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.

Q: How does the CLT when dimension $d$ grows too fast relate to other results in probability theory?

A: The CLT when dimension $d$ grows too fast is related to other results in probability theory, such as the law of large numbers (LLN) and the Berry-Esseen theorem. The LLN states that the average of a large number of iid random variables converges to the population mean, while the Berry-Esseen theorem provides a bound on the rate of convergence of the CLT.

Q: What are some applications of the CLT when dimension $d$ grows too fast?

A: Some applications of the CLT when dimension $d$ grows too fast include signal processing, image analysis, and machine learning. In these fields, high-dimensional data is common, and the CLT when dimension $d$ grows too fast provides a way to understand the behavior of the sum of iid random variables.

Q: What are some open problems related to the CLT when dimension $d$ grows too fast?

A: Some open problems related to the CLT when dimension $d$ grows too fast include:

• Developing new results and techniques to understand the behavior of the sum of iid random variables in high dimensions.
• Investigating the rate of convergence of the CLT when dimension $d$ grows too fast.
• Applying the CLT when dimension $d$ grows too fast to real-world problems, such as signal processing, image analysis, and machine learning.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the CLT when dimension $d$ grows too fast. We have discussed the importance of the CLT, the condition for the CLT to hold when dimension $d$ grows too fast, and the distribution of the sum of iid random variables when dimension $d$ grows too fast. We have also discussed some applications and open problems related to the CLT when dimension $d$ grows too fast.

References

• [1] Bhattacharya, R. N. (1987). Asymptotic normality and the law of the iterated logarithm for the sample covariance matrix. The Annals of Probability, 15(2), 347-362.
• [2] Götze, F. (1989). On the asymptotic distribution of the sample covariance matrix. Journal of Multivariate Analysis, 29(2), 145-155.
• [3] Paulauskas, V. (1990). On the asymptotic behavior of the sample covariance matrix. Journal of Multivariate Analysis, 33(2), 147-155.

Appendix

In this appendix, we provide some additional information related to the CLT when dimension $d$ grows too fast.

• The CLT when dimension $d$ grows too fast is a special case of the CLT. The CLT when dimension $d$ grows too fast is a special case of the CLT, which states that the distribution of the sum of iid random variables converges to a normal distribution with mean vector $\mu$ and variance-covariance matrix $\Sigma$.
• The CLT when dimension $d$ grows too fast is related to the law of large numbers (LLN). The CLT when dimension $d$ grows too fast is related to the law of large numbers (LLN), which states that the average of a large number of iid random variables converges to the population mean.
• The CLT when dimension $d$ grows too fast is related to the Berry-Esseen theorem. The CLT when dimension $d$ grows too fast is related to the Berry-Esseen theorem, which provides a bound on the rate of convergence of the CLT.