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The question asks to show that if \( X_n \) possesses a **t-distribution** with \( n \) degrees of freedom, then as \( n \to \infty \), \( X_n \) converges in distribution to a **standard normal distribution**, i.e., \( X_n \overset{d}{\longrightarrow} N(0, 1) \).

### Approach:

1. **t-distribution Definition**: The t-distribution with \( n \) degrees of freedom can be represented as:

   \[
   X_n = \frac{Z}{\sqrt{\frac{V_n}{n}}}
   \]
   
   where:
   - \( Z \sim N(0, 1) \) is a standard normal random variable.
   - \( V_n \sim \chi^2_n \) is a chi-squared random variable with \( n \) degrees of freedom, independent of \( Z \).

2. **Limiting Behavior of \( V_n \)}: As \( n \to \infty \),
   - \( \frac{V_n}{n} \) converges in probability to 1, by the Law of Large Numbers (since \( V_n \) is the sum of \( n \) independent \( \chi^2_1 \) random variables, and its expected value is \( n \)).

   Formally:
   \[
   \frac{V_n}{n} \overset{P}{\longrightarrow} 1 \quad \text{as} \quad n \to \infty.
   \]

3. **Substituting into \( X_n \)**: Therefore, as \( n \to \infty \),
   \[
   X_n = \frac{Z}{\sqrt{\frac{V_n}{n}}} \overset{d}{\longrightarrow} \frac{Z}{\sqrt{1}} = Z.
   \]
   Since \( Z \sim N(0, 1) \), this shows that \( X_n \) converges in distribution to \( N(0, 1) \).

Thus, we have demonstrated that \( X_n \overset{d}{\longrightarrow} N(0, 1) \) as \( n \to \infty \).

### Conclusion:

The limiting distribution of the t-distribution with \( n \) degrees of freedom converges to the standard normal distribution \( N(0, 1) \) as \( n \to \infty \), based on the behavior of the chi-squared random variable and its relationship to the t-distribution.

Would you like more details on any part of this explanation?

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Here are 5 related questions:
1. What is the intuition behind the Central Limit Theorem in this context?
2. How does the chi-squared distribution behave as degrees of freedom increase?
3. Can we generalize this result for other types of distributions?
4. How does this result apply to hypothesis testing involving large samples?
5. What is the relationship between the t-distribution and normal distribution for small \(n\)?

**Tip**: Understanding limiting distributions helps in simplifying statistical analysis for large samples, especially when dealing with hypothesis testing or confidence intervals!

If X_n possesses a t-distribution with n degrees of freedom, then X_n converges to N(0, 1) as n → ∞. Show this.

This solution demonstrates that a t-distribution with n degrees of freedom converges to the standard normal distribution N(0, 1) as n → ∞, using the law of large numbers and properties of chi-squared random variables.

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