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?
The problem asks to identify the correct statements about the convergence of a scaled sum of independent and identically distributed (i.i.d.) random variables, $Y_n = X_1 + \cdots + X_n$. The random variable $X_1$ has the probability mass function $P(X_1 = 1) = P(X_1 = 3) = \frac{1}{2}$.
Calculating Properties of X1
First, we determine the expectation and variance of $X_1$.
- Expectation ($E[X_1]$): $ E[X_1] = \sum x P(X_1 = x) = (1 \times \frac{1}{2}) + (3 \times \frac{1}{2}) = \frac{1+3}{2} = 2 $
- Second Moment ($E[X_1^2]$): $ E[X_1^2] = \sum x^2 P(X_1 = x) = (1^2 \times \frac{1}{2}) + (3^2 \times \frac{1}{2}) = \frac{1+9}{2} = 5 $
- Variance ($Var(X_1)$): $ Var(X_1) = E[X_1^2] - (E[X_1])^2 = 5 - 2^2 = 5 - 4 = 1 $
Analyzing Convergence Statements
We analyze each statement using the properties of i.i.d. random variables, the Weak Law of Large Numbers (WLLN), and variance calculations.
Statement A: $\frac{Y_n}{n}$ converges to $2$ in probability.
The term $\frac{Y_n}{n}$ represents the sample mean of the first $n$ i.i.d. random variables. The WLLN states that the sample mean converges in probability to the expected value, provided the expectation is finite.
Since $E[X_1] = 2$, by WLLN:
$ \frac{Y_n}{n} \xrightarrow{P} E[X_1] = 2 $Therefore, Statement A is correct.
Statement B: $\text{Variance}\left(\frac{Y_n}{n^{2/3}}\right)$ converges to $0$, as $n \to \infty$.
For the sum $Y_n$ of $n$ i.i.d. random variables:
$ Var(Y_n) = n \times Var(X_1) = n \times 1 = n $Now, we find the variance of the scaled variable:
$ \text{Var}\left(\frac{Y_n}{n^{2/3}}\right) = \left(\frac{1}{n^{2/3}}\right)^2 Var(Y_n) = \frac{1}{n^{4/3}} \times n = \frac{n}{n^{4/3}} = \frac{1}{n^{1/3}} $As $n \to \infty$, $\frac{1}{n^{1/3}} \to 0$.
Therefore, Statement B is correct.
Statement C: $\frac{Y_n}{n^{2/3}}$ converges to $c$ in probability, where $0 < c < \infty$.
From WLLN (Statement A), $\frac{Y_n}{n} \to 2$ in probability. This suggests $Y_n \approx 2n$ for large $n$. Consider the scaling factor $n^{2/3}$:
$ \frac{Y_n}{n^{2/3}} \approx \frac{2n}{n^{2/3}} = 2n^{1 - 2/3} = 2n^{1/3} $As $n \to \infty$, $2n^{1/3} \to \infty$. The expression does not converge to a finite constant $c$.
Therefore, Statement C is incorrect.
Statement D: $\frac{Y_n}{n^2}$ converges to $0$ in probability.
We can rewrite the expression as:
$ \frac{Y_n}{n^2} = \left(\frac{Y_n}{n}\right) \times \left(\frac{1}{n}\right) $We know $\frac{Y_n}{n} \xrightarrow{P} 2$ and $\frac{1}{n} \to 0$ as $n \to \infty$. The product of a sequence converging in probability to a constant and a sequence converging to 0 converges to 0.
Therefore, $\frac{Y_n}{n^2} \xrightarrow{P} 2 \times 0 = 0$. Statement D is correct.
Summary of Correct Statements
The statements found to be correct are A, B, and D.
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 $\frac{1}{\sqrt{2\pi}} \int_2^\infty e^{-x^2/2} dx$?
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,
$$\lim_{n\to\infty} P(X_1 + X_2 + \cdots + X_n > n \log n)$$
