In this problem, we have a sequence of independent and identically distributed random variables \(X_1, X_2, \ldots\) with specified moments: \(E(X_1) = 0\), \(E(X_1^2) = 1\), \(E(X_1^3) = 0\), and \(E(X_1^4) = 3\). We need to analyze the given statements about convergences of different sequences.
- Statement 1: \(\frac{S_n}{\sqrt{n}}\) converges in distribution to a random variable \(Z\), where \(Z \sim N(0, 1)\).
This is a classic application of the Central Limit Theorem (CLT). As \(S_n = \sum_{i=1}^n X_i\), where each \(X_i\) is i.i.d with mean 0 and variance 1, by CLT, \(\frac{S_n}{\sqrt{n}}\) converges in distribution to a standard normal distribution, \(N(0, 1)\).
This statement is True. - Statement 2: \(\frac{T_n-n}{\sqrt{3n}}\) converges in distribution to a random variable \(Z\), where \(Z \sim N(0,1)\).
To analyze this, notice that \(T_n = \sum_{i=1}^n X_i^2\), and we know \(E(X_i^2) = 1\). By the Law of Large Numbers, \(\frac{T_n}{n} \to 1\). For the convergence in distribution, we need the variance, which is not \(3n\) (as variance of sum of squares is not \(3n\)).
This statement is False. - Statement 3: \(\frac{\sqrt{n}S_n}{T_n}\) converges in distribution to a random variable \(Z\), where \(Z \sim N(0, 1)\).
Since \(S_n\) follows a normal distribution and \(T_n/n \to 1\) in probability, this transforms \(\frac{\sqrt{n}S_n}{T_n}\) to a form that resembles a normalized sum. However, more context would be needed to verify convergence due to dependence on \(T_n\)'s behavior, but an assumption for a random variable quotient tends to standardize variance.
This statement can be considered True. - Statement 4: \(\frac{T_n-n}{\sqrt{V_n}}\) converges in distribution to a random variable \(Z\), where \(Z \sim N(0,1)\).
Similar reasoning as in Statement 2; however, \(V_n\)
Therefore, the correct answers are Statements 1 and 3.
Important Questions from Central Limit Theorems
- 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
- Let $X_i$'s be independent random variables such that $X_i$'s are symmetric about 0 and $\text{Var}(X_i) = 2i-1$, for $i \ge 1$. Then,
$$\lim_{n\to\infty} P(X_1 + X_2 + \cdots + X_n > n \log n)$$
