The Central Limit Theorem

Suppose the CLT holds for a sequence $\{ \xi_{n}\}$, and $\operatorname {Var}\left[S_{n}\right] = o(n^{2})$ as $n \rightarrow \infty$. Prove that the LLN then holds as well.

Give an example showing that the Lyapunov condition is not necessary for the CLT to hold.

Let $\xi_{1}, \xi_{2}, \ldots$ be a uniformly bounded sequence of independent random variables, i.e., there exists a constant $A < \infty$ such that $\left|\xi_{n}\right| < A$ for all $n$. Let $S_{n} = \xi_{1} + \ldots + \xi_{n}$. Prove that the sequence $\{ \xi_{n}\}$ satisfies the CLT if $\operatorname {Var}\left[S_{n}\right] \rightarrow \infty$.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite, nonzero variance. Prove that for any $a$ and $b$

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite variance. Prove that for any $b$ the limit

equals either 0, 1, or 1/2. Find the conditions under which each of these situations occurs.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed (nondegenerate) random variables with zero mean and finite variance. Prove that for any $b > 0$ the limit

equals 0 for $\alpha < 1/2$ and 1 for $\alpha > 1/2$.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed (nondegenerate) random variables with zero mean and finite variance. Find the limit

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero mean and finite variance. Find $\operatorname {Var}\left[\xi_{1}\right]$ if:

The random variable $\eta_{n}$ has a Poisson distribution with parameter $n$. Does the limit as $n \rightarrow \infty$ of the ratio $(\eta_{n} - n)/\sqrt{n}$ exist?

Prove that as $n \rightarrow \infty$

The random variable $\eta_{n}$ has a binomial distribution with parameters $n$ and 1/2. Does the limit as $n \rightarrow \infty$ of the ratio $(2\eta_{n} - n)/\sqrt{n}$ exist?

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $n$ and $-n$ with equal probabilities. Does the distribution of the ratio converge to a normal law:

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $n$, $-n$, and 0 with probabilities:

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $\sqrt{n}$, $-\sqrt{n}$, and 0 with probabilities $1/2n$, $1/2n$, and $1 - 1/n$ respectively. Does the CLT hold for the sequence $\{ \xi_{n}\}$?

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $2^{n}$ and $-2^{n}$ with probability $1/2$ each. Does the CLT hold for the sequence $\{ \xi_{n}\}$?

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $2^{n}$, $-2^{n}$, and 0 with probabilities $2^{-(2n+1)}$, $2^{-(2n+1)}$, and $1 - 2^{-2n}$ respectively. Does the CLT hold for the sequence $\{ \xi_{n}\}$?

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ be uniformly distributed on the segment $[a_{n} - 1, a_{n} + 1]$, where $\sum_{n} \left|a_{n}\right| < \infty$. Find

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values 1 and $-1$ with probability $(1 - 2^{-n})/2$ each, and the values $2^{n}$ and $-2^{n}$ with probability $2^{-n-1}$ each. Prove that the CLT holds for the sequence $\{ \xi_{n}\}$.

Let $\xi_{1}, \xi_{2}, \ldots$ have a Poisson distribution with parameter $\lambda$. Does the weak limit as $n \rightarrow \infty$ of the expression

exist?

Let $\xi$ and $\eta$ be independent identically distributed random variables with finite variance. Prove that if the variables $\xi + \eta$ and $\xi - \eta$ are independent, then $\xi$ and $\eta$ have a normal distribution, possibly degenerate.

Let $\xi$ and $\eta$ be independent identically distributed random variables with zero mean and finite variance. Using the CLT, prove that if $\xi$ and $(\xi + \eta )/\sqrt{2}$ have the same nondegenerate distribution, then that distribution is normal.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Let $\xi_{n}$ take the values $a_{n}$, $-a_{n}$, and 0 with probabilities $p_{n}$, $p_{n}$, and $1 - 2p_{n}$ respectively. Find conditions on the constants $a_{n}$ and $p_{n}$ that ensure the CLT holds for the sequence $\{ \xi_{n}\}$.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. When $n$ is a perfect square, $\xi_{n}$ takes the values $-\sqrt{n}$ and $\sqrt{n}$ with probability $1/2$ each; for all other $n$, $\xi_{n}$ takes the values $-1$ and 1 with probability $1/2$ each. Does the CLT hold for the sequence $\{ \xi_{n}\}$?

Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random variables with $\mathbb {E}\left[\xi_{1}\right] = 0$ and $\operatorname {Var}\left[\xi_{1}\right] = 1$. Find the limiting distribution of the random variables

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with $\mathbb {E}\left[\xi_{1}\right] = 0$ and $\operatorname {Var}\left[\xi_{1}\right] = \sigma^{2} > 0$. Do the quantities

converge in probability as $n \rightarrow \infty$?