Thank you for the correction. Let's re-evaluate the problem with the correct expression for $X_i$.

Given the random variables $X_i$ with the distribution defined as:

$X_i = 2^k - 2\log(k) \quad \text{with probability} \quad 2^{-k}, \quad k = 1, 2, 3, \dots$

We need to determine if the Law of Large Numbers (LLN) holds for this sequence $\{X_i\}$ of independent and identically distributed (i.i.d.) random variables. To do this, we'll evaluate the expected value $\mathbb{E}[X_i]$.

Step 1: Calculate the Expected Value $\mathbb{E}[X_i]$

The expected value $\mathbb{E}[X_i]$ is given by:

$\mathbb{E}[X_i] = \sum_{k=1}^{\infty} \left(2^k - 2\log(k)\right) \cdot 2^{-k}$

This expression can be split into two sums:

$\mathbb{E}[X_i] = \sum_{k=1}^{\infty} 2^k \cdot 2^{-k} - 2\sum_{k=1}^{\infty} \log(k) \cdot 2^{-k}$

Step 2: Evaluate the First Sum

The first sum simplifies as follows:

$\sum_{k=1}^{\infty} 2^k \cdot 2^{-k} = \sum_{k=1}^{\infty} 1 = \infty$

This sum diverges because we are summing 1 infinitely many times.

Step 3: Evaluate the Second Sum

The second sum is:

$-2\sum_{k=1}^{\infty} \log(k) \cdot 2^{-k}$

This sum converges to a finite value because $\log(k)$ grows much slower than $2^k$ decreases, so the weighted sum with the probability $2^{-k}$ converges.

Step 4: Conclusion on the Expected Value

Since the first part of the expected value sum diverges (i.e., it is infinite), the expected value $\mathbb{E}[X_i]$ does not exist or is infinite.

Step 5: Implication for the Law of Large Numbers

For the Law of Large Numbers to hold, the expected value $\mathbb{E}[X_i]$ must be finite. Since $\mathbb{E}[X_i]$ is infinite, the Law of Large Numbers does not hold for this sequence $\{X_i\}$. This means the sample average will not converge to a finite expected value as the number of observations increases.

Summary

The Law of Large Numbers does not hold for the sequence $\{X_i\}$ because the expected value $\mathbb{E}[X_i]$ is infinite.

Would you like more details on why the first sum diverges or have any other questions?

Follow-up Questions

1. What are the implications of a divergent expected value in probability theory?
2. Can you provide an example where the expected value is finite, and the Law of Large Numbers holds?
3. How does one determine the convergence or divergence of a series in a distribution?
4. What are the consequences of the Law of Large Numbers not holding?
5. Could you explain how different distributions affect the applicability of the Law of Large Numbers?

Tip

When evaluating the expected value for the application of the Law of Large Numbers, carefully check the behavior of each term in the sum. A divergent sum indicates that the Law of Large Numbers may not hold.