Part VI

Limit Theorems

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Section 25The Central Limit Theorem[25/64%]

Problem 25.1

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.

Problem 25.2

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

Problem 25.3

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 $\operatorname {Var}\left[S_{n}\right] \rightarrow \infty$ .

Problem 25.4

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 $b$

\lim _{n \rightarrow \infty } \mathbb {P}\left(a \leqslant S_{n} \leqslant b\right) = 0.

Problem 25.5

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

\lim _{n \rightarrow \infty } \mathbb {P}\left(S_{n} < b\right)

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

Problem 25.6

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

\lim _{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{S_{n}}{n^{\alpha }}\right| < b\right)

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

Problem 25.7

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

\lim _{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{S_{n}}{\sqrt{n}}\right| < b\right).

Problem 25.8

(a)(b)

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:

(a)

$\lim_{n \rightarrow \infty } \mathbb {P}\left(\frac{S_{n}}{\sqrt{n}} > 1\right) = \frac{1}{3}$ ;

(b)

$\lim_{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{S_{n}}{\sqrt{n}}\right| > 1\right) = \frac{1}{5}$ .

Problem 25.9

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?

Problem 25.10

Prove that as $n \rightarrow \infty$

e^{-n} \sum _{k=0}^{n} \frac{n^{k}}{k!} \rightarrow \frac{1}{2}.

Problem 25.11

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?

Problem 25.12

(a)(b)

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:

(a)

$S_{n}/\sqrt{n}$ ;

(b)

$S_{n}/\sqrt{\operatorname {Var}\left[S_{n}\right]}$ ?

Problem 25.13

(a)(b)(c)(d)

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:

(a)

$1/2n^{2}$ , $1/2n^{2}$ , and $1 - 1/n^{2}$ ;

(b)

$1/2\sqrt{n}$ , $1/2\sqrt{n}$ , and $1 - 1/\sqrt{n}$ ;

(c)

$1/4$ , $1/4$ , and $1/2$ ;

(d)

$2^{-n}$ , $2^{-n}$ , and $1 - 2^{-n+1}$ .

Does the CLT hold for the sequence $\{ \xi_{n}\}$ ?

Problem 25.14

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 $\{ \xi_{n}\}$ ?

Problem 25.15

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 $\{ \xi_{n}\}$ ?

Problem 25.16

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 $\{ \xi_{n}\}$ ?

Problem 25.17

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

\lim _{n \rightarrow \infty } \mathbb {P}\left(0 < \frac{S_{n}}{\sqrt{n}} < 1\right).

Problem 25.18

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 $\{ \xi_{n}\}$ .

Problem 25.19

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

\frac{1}{\sqrt{n}} \sum _{i=1}^{n} (\xi _{2i-1} - \xi _{2i})?

exist?

Problem 25.20

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.

Problem 25.21

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.

Problem 25.22

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 $\{ \xi_{n}\}$ .

Problem 25.23

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 $\{ \xi_{n}\}$ ?

Problem 25.24

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

\eta _{n} = \sqrt{n} \frac{\xi _{1} + \ldots + \xi _{n}}{\xi _{1}^{2} + \ldots + \xi _{n}^{2}} \quad \text{and} \quad \zeta _{n} = \frac{\xi _{1} + \ldots + \xi _{n}}{\sqrt{\xi _{1}^{2} + \ldots + \xi _{n}^{2}}}.

Problem 25.25

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

\frac{\xi _{1} + \ldots + \xi _{n}}{\sqrt{n}}?

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