## Example Problems on Limit Theorems

ST501 Homework 7 on Limit Theorems

Problems: 5.2, 5.5, 5.6, 5.19, 5.28, Extra Problem 1

# 5.2

Let $X_i$ be as in Problem 1 but with $E(X_i) = \mu_i$ and $n^{-1}\sum_{i=1}^{n}\mu_i \rightarrow \mu$. Show that $\bar X \rightarrow \mu$ in probability.

Since these distributions have different means, we cannot apply the Law of Large Numbers. Thus, we will start this the definition of convergence in probability.

\begin{align} \lim_{n\rightarrow \infty} P\Big[ | \bar{X_n} - \mu | \geq \epsilon \Big] & = \lim_{n\rightarrow \infty} P\Big[(\bar{X_n} - \mu)^2 \geq \epsilon^2 \Big] \\ & \leq \lim_{n\rightarrow \infty} \frac{E\Big[ (\bar{X_n} - \mu )^2 \Big]}{\epsilon^2} & \text{Markov's Inequality} \end{align}

Let’s look at the numerator,

\begin{align} E\Big[ (\bar{X_n} - \mu )^2 \Big] & = E\Big[ (\bar{X_n} - \mu + E(\bar{X_n}) - \bar{X_n})^2 \Big] & = \text{Adding } (E(\bar{X_n}) - E(\bar{X_n}))\\ & = E\Big[ \Big( (\bar{X_n} - E(\bar{X_n})) + (E(\bar{X_n})- \mu ) \Big)^2 \Big] \\ & = E\Big[ (\bar{X_n} - E(\bar{X_n}))^2 + 2(\bar{X_n} - E(\bar{X_n}))(E(\bar{X_n})- \mu ) + (E(\bar{X_n})- \mu )^2 \Big] \\ & = E\Big[ (\bar{X_n} - E(\bar{X_n}))^2 \Big]+ E\Big[2(\bar{X_n} - E(\bar{X_n}))(E(\bar{X_n})- \mu ) \Big]+ E\Big[(E(\bar{X_n})- \mu )^2 \Big] & E \text{ is a linear operator} \\ & = Var(\bar{X_n}) + 0 + (E(X_n) - \mu)^2 & \text{simplifying Expected values} \\ & = Var(\frac{1}{n} \sum_{i=1}^{n}X_i) + (\frac{1}{n}\sum_{i=1}^{n} \mu_i - \mu)^2 \\ & = \frac{1}{n^2} \sum_{i=1}^{n}\sigma_{i} + (\frac{1}{n}\sum_{i=1}^{n} \mu_i - \mu)^2 \end{align}

Now we can back to the original inequality and take the limit.

\begin{align} \lim_{n\rightarrow \infty} \frac{E\Big[ (\bar{X_n} - \mu )^2 \Big]}{\epsilon^2} & = \frac{\frac{1}{n^2} \sum_{i=1}^{n}\sigma_{i} + (\frac{1}{n}\sum_{i=1}^{n} \mu_i - \mu)^2}{\epsilon^2} \\ & = \frac{0 + (\mu - \mu)}{\epsilon^2} & \text{Using limits from the problem} \\ & = 0 \end{align}

Thus, by definition, $\bar X \xrightarrow{p} \mu$.

# 5.5

Using moment-generating functions, show that as $n\rightarrow \infty, \ p \rightarrow 0$, and $np\rightarrow \lambda$, the binomial distribution with parameters $n$ and $p$ tends to the Poisson distribution.

To show convergence in distribution with MGFs, we can use the definition: $Y_n \xrightarrow{d} Y$ if $\lim_{n\rightarrow \infty} M_{Y_n}(y) = M_Y(y)$.

\begin{align} M_{Y_n}(y) & = (p e^t + (1-p)^n \\ & = (p e^t + 1 - p)^n \\ & = (1 + p(e^t - 1))^n \\ & = (1 + \frac{n}{n} p(e^t - 1))^n \\ & = (1 + \frac{np}{n}(e^t - 1))^n \end{align}

Now we can take the limits.

\begin{align} \lim_{np\rightarrow \lambda} M_{Y_n}(y) & = (1+\frac{\lambda}{n}(e^t - 1))^n \\ & = \lim_{n\rightarrow \infty} (1+\frac{\lambda}{n}(e^t - 1))^n \\ & = e^{\lambda(e^t - 1)} \end{align}

Notice that this is the MGF of a Poisson distribution. Thus, $Bin(n,p) \xrightarrow{d} Poi(\lambda)$ with $n\rightarrow \infty, \ p \rightarrow 0$, and $np\rightarrow \lambda$.

# 5.6

Using moment-generating functions, show that as $\alpha \rightarrow \infty$ the gamma distribution with parameters $\alpha$ and $\lambda$, properly standardized, tends to the standard normal distributions

We will need to recall some properties of the Gamma distribution.

\begin{align} X_i\sim Gamma(\alpha_i, \lambda) & \rightarrow \frac{1}{n} \sum_{i=1}^{n}X_i \sim Gamma(\sum_{i=1}^{n} \alpha_i, \lambda)\\ Y\sim Gamma(\alpha, \lambda) & \rightarrow cY\sim Gamma(\alpha, c \lambda) \end{align}

Take $Y \sim Gamma(\alpha, \lambda)$. With this, we can write $X_i \sim Gamma(\alpha,\frac{\lambda}{n})$ so that $Y_n=\frac{1}{n} \sum_{i=1}^{n} X_i$. Standardizing this gives

$W = \frac{Y - E(Y)}{\sqrt{Var(Y)}}.$

Next we can examine the MGF of $W$.

\begin{align} M_W(t) & = E\Bigg[ \exp(\frac{Y - \frac{\alpha}{\lambda}}{\sqrt{\frac{\alpha}{\lambda^2}}}t) \Bigg]\\ & = e^{-\sqrt{\alpha}t}\cdot E\Bigg[ e^{\frac{\lambda y}{\sqrt{\alpha}}t} \Bigg] \\ \end{align}

Notice that the expectation value follows the form of a gamma MGF with parameter $\frac{\lambda}{\alpha}t$. We can substitute in the MGF.

$M_W(t) = e^{-\sqrt{\alpha}t}\cdot \Big( 1 - \frac{t}{\sqrt{\alpha}} \Big)^{-\alpha}$

To simplify this expression, we can take the $\log$.

$\log(M_W(t)) = -\sqrt{\alpha} t - \alpha \log(1-\frac{t}{\sqrt{\alpha}})$

From this we can look at the Taylor series expansion to the second term of the logarithm.

$\log(1-x) = x-\frac{x^2}{2} + \dots$ \begin{align} \log(M_W(t)) & = -\sqrt{\alpha} t - \alpha \Big( \frac{t}{\sqrt{\alpha}} - \frac{t^2}{2\alpha} \Big) \\ & = -\sqrt{\alpha} t + \sqrt{\alpha} t + \frac{t^2}{2} \end{align}

Now we can go back and exponentiate this to get back to our original MGF.

$M_W(t) = e^{\frac{t^2}{2}}$

Notice that this is the MGF of a standard normal distribution. Thus, by MGFS, $W \xrightarrow{d} Z$.

# 5.19

## a.

Use the Monte Carlo method with $n = 100$ and $n = 1000$ to estimate $\int_0^1 \cos(2\pi x)dx$. Compare the estimates to the exact answer.

See R Code.

## b.

Use Monte Carlo to evaluate $\int_0^1 \cos(2\pi x^2)dx$. Can you find the exact answer?

See R Code.

# 5.28

Let $f_n$ be a sequence of frequency functions with $f_n(x) = \frac{ 1 }{ 2 }$ if $x = \pm \Big( \frac{ 1 }{2} \Big)^n$ and $f_n(x) = 0$ otherwise. Show that $\lim f_n(x) = 0$ for all $x$ which means that the frequency functions do not converge to a frequency function, but there exists a CDF $F$ such that $\lim F_n(x) = F(x)$.

Notice that

$f_n(x) = \begin{cases} \frac{1}{2} & \text{for } x = \pm \Big( \frac{ 1 }{2} \Big)^n\\ 0 & \text{otherwise.} \end{cases}$

This function goes to 0 pointwise as n goes to infinity, $\lim_{n\rightarrow \infty} f_n = 0$. This is not a valid PDF. However, the limit of the CDF does converge to a valid CDF function.

$\lim_{n\rightarrow \infty} F_n(x) \begin{cases} 0 & x < 0 1 & x \geq 0 \end{cases}$

This is an example of why we cannot use PDFs to show convergence as the results do not always match those of the proven CDF.

# Extra Problem 1

Suppose $X_i$ are independent random variables and that $E(X_i)=\mu, \ Var(X_i) = \sigma^2$.

## a.

What is the mean of $(\bar X)^2$?

$E((\bar X)^2) = \mu^2$

## b.

Prove that $(\bar X)^2$ converges in probability to a constant and give that constant. State any theorems or results you use in your proof.

Since each $X_i$ is independent and has the same mean and variance, we can apply the Weak Law of Large Numbers to see that

$\bar{X_n} \xrightarrow{p} \mu.$

Using the continuity theorem, we get

$E\Big( (\bar{X_n})^2 \Big) \xrightarrow{p} \mu^2.$