To analyze the given statements, we need to understand the behavior of the sum of squares of i.i.d. standard normal random variables. Here, \(S_n = X_1^2 + X_2^2 + \cdots + X_n^2\) is a sum of squares of i.i.d. \(N(0, 1)\) random variables. Each \(X_i^2\) follows a chi-squared distribution with 1 degree of freedom.
Let's evaluate each statement individually:
Statement: \(\frac{S_n - n}{\sqrt{2}} \sim N(0, 1)\) for all \(n \ge 1\).
Explanation: \(S_n\) is a chi-squared distribution with \(n\) degrees of freedom. Generally, \(\frac{S_n - n}{\sqrt{2n}}\) converges in distribution to \(N(0, 1)\) by the Central Limit Theorem for chi-square distributions as \(n\) increases, but not \(\frac{S_n - n}{\sqrt{2}}\). Hence, this statement is incorrect.
Statement: For all \(\varepsilon > 0\), \(P\left(\left|\frac{S_n}{n} - 2\right| > \varepsilon\right) \to 0\) as \(n \to \infty\).
Explanation: According to the Law of Large Numbers, \(\frac{S_n}{n}\) converges to the expected value of \(X_i^2\), which is 1 (not 2). Therefore, this statement is incorrect.
Statement: \(\frac{S_n}{n} \to 1\) with probability 1.
Explanation: By the Strong Law of Large Numbers, as \(n \to \infty\), the average of \(X_i^2\) converges almost surely to its expected value, which is 1. Thus, this statement is correct.
Statement: \(P(S_n \le n + \sqrt{n} x) \to P(Y \le x)\) for all \(x \in \mathbb{R}\), where \(Y \sim N(0, 2)\).
Explanation: Applying the Central Limit Theorem for chi-square distributions, \(\frac{S_n - n}{\sqrt{2n}}\) converges to \(N(0, 1)\). Scaling and shifting gives \(S_n \le n + \sqrt{n} x\) converging to \(N(0, 2)\). Therefore, this statement is correct.
Conclusion: The correct statements are \(\frac{S_n}{n} \to 1\) with probability 1 and \(P(S_n \le n + \sqrt{n} x) \to P(Y \le x)\) where \(Y \sim N(0, 2)\).
Important Questions from Central Limit Theorems
- Let $X_1, X_2, . . .$ be a sequence of independent and identically distributed random variables with $E(X_1) = 0, E(X_1^2) = 1, E(X_1^3) = 0, E(X_1^4) = 3$. Let $S_n = \sum_{i=1}^n X_i, T_n = \sum_{i=1}^n X_i^2, U_n = \sum_{i=1}^n X_i^3$ and $V_n = \sum_{i=1}^n X_i^4$. Then, which of the following statements are true?
- Let $\{X_i; i \ge 1\}$ be a sequence of independent random variables each having a normal distribution with mean 2 and variance 5. Then which of the following are true
- For $n \ge 1$, let $X_n$ be a Poisson random variable with mean $n^2$. Which of the following are equal to $\frac{1}{\sqrt{2\pi}} \int_2^\infty e^{-x^2/2} dx$?
Let $\{X_n : n \ge 1\}$ be a sequence of independent and identically distributed random variables and the probability mass function of $X_1$ is the following;
$P(X_1 = 1) = P(X_1 = 3) = \frac{1}{2}.$ If $Y_n = X_1 + \cdots + X_n$,
then which of the following statements are correct?
Suppose $X_1, X_2, \dots$ are independent random variables. Assume that $X_1, X_3, \dots$ are identically distributed with mean $\mu_1$ and variance $\sigma_1^2$, while $X_2, X_4, \dots$ are identically distributed with mean $\mu_2$ variance $\sigma_2^2$. Let $S_n = X_1 + X_2 + \dots + X_n$. Then $\frac{S_n - a_n}{b_n}$ converges in distribution to $N(0,1)$ if
