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An Intermediate Course in Probability

Allan Gut

Chapter 6

Convergence - all with Video Answers

Educators

Chapter Questions

06:21

Problem 1

Let $\left\{X_{n}, n \geq 1\right\}$ be a sequence of independent, identically distributed random variables with density
$$
f(x)= \begin{cases}e^{-(x-a)}, & \text { for } x \geq a, \\ 0, & \text { for } x<a .\end{cases}
$$
Set $Y_{n}=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}$. Show that
$$
Y_{n} \stackrel{p}{\longrightarrow} a \text { as } n \rightarrow \infty .
$$
Remark. This is a translated exponential distribution; technically, if $X$ is distributed as above, then $X-a \in \operatorname{Exp}(1)$. We may interpret this as $X$ having age $a$ and a remaining lifetime $X-a$, which is standard exponential.

Mengchun Cai
Mengchun Cai
Numerade Educator
02:36

Problem 2

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed random variables such that $P\left(X_{k} \leq a\right)=1$ for some real $a$. Show that
$$
\max _{1 \leq k \leq n} X_{k} \stackrel{p}{\longrightarrow} a \text { as } n \rightarrow \infty .
$$
Remark. $a$ is the smallest possible number such that $P\left(X_{k} \leq a\right)=1$.

Amany Waheeb
Amany Waheeb
Numerade Educator
16:48

Problem 3

Let $X_{1}, X_{2}, \ldots$ be independent, $C(0,1)$-distributed random variables. Determine the limit distribution of
$$
Y_{n}=\frac{1}{n} \cdot \max \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}
$$
as $n \rightarrow \infty$.
Remark. It may be helpful to know that $\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}$ and that arctan $y=y-y^{3} / 3+y^{5} / 5-y^{7} / 7+\ldots .$

Mengchun Cai
Mengchun Cai
Numerade Educator
16:48

Problem 4

Suppose that $X_{1}, X_{2}, \ldots$ are independent, $\mathrm{Pa}(1,2)$-distributed random variables, and set $Y_{n}=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\} .$
(a) Show that $Y_{n} \stackrel{p}{\longrightarrow} 1$ as $n \rightarrow \infty$.
It thus follows that $Y_{n} \approx 1$ with a probability close to 1 when $n$ is large. One might therefore suspect that there exists a limit theorem to the effect that $Y_{n}-1$, suitably rescaled, converges in distribution as $n \rightarrow \infty$ (note that $Y_{n}>1$ always).
(b) Show that $n\left(Y_{n}-1\right)$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Mengchun Cai
Mengchun Cai
Numerade Educator
16:48

Problem 5

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed, continuous random variables. Extreme value theory is concerned with the limiting behavior of $Y_{n}=\max \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}$ (suitably normalized) as $n \rightarrow \infty$. One can show that precisely three different limit distributions may occur. One of them is the distribution whose distribution function is given by
$$
\Lambda(x)=e^{-e^{-s}}, \text { for }-\infty<x<\infty .
$$
Show that the exponential distribution belongs to this class. More precisely, show that if $X_{1}, X_{2}, \ldots \in \operatorname{Exp}(1)$, then
$$
Y_{n}-\log n \stackrel{d}{\longrightarrow} \Lambda \text { as } n \rightarrow \infty
$$

Mengchun Cai
Mengchun Cai
Numerade Educator
01:10

Problem 6

Let $X_{1}, X_{2}, \ldots$ be as given in the previous problem. Another, socalled extreme value distribution has distribution function
$$
G(x)=e^{-x^{-\alpha}}, \text { for } x>0
$$
for some $\alpha>0$ (and $G(x)=0$ when $x<0$ ). Show that the Pareto distribution belongs to this class. More precisely, suppose that $X_{1}, X_{2}, \ldots$ are independent, $\mathrm{Pa}(\beta, \alpha)$-distributed random variables, and set $Y_{n}=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\} .$ Show that
$$
\frac{Y_{n}}{\beta n^{1 / \alpha}} \stackrel{d}{\longrightarrow} G \quad \text { as } \quad n \rightarrow \infty .
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
03:04

Problem 7

Suppose that $X_{n} \in \operatorname{Ge}\left(\frac{\lambda}{n+\lambda}\right), n=1,2, \ldots$, where $\lambda$ is a positive constant. Show that $X_{n} / n$ converges in distribution to an exponential distribution as $n \rightarrow \infty$, and determine the parameter of the limit distribution.

Narayan Hari
Narayan Hari
Numerade Educator
07:49

Problem 8

Let $X_{1}, X_{2}, \ldots$ be a sequence of random variables such that
$$
P\left(X_{n}=\frac{k}{n}\right)=\frac{1}{n}, \quad \text { for } \quad k=1,2, \ldots, n .
$$
Determine the limit distribution of $X_{n}$ as $n \rightarrow \infty$.

Abhirup Pal
Abhirup Pal
Numerade Educator
07:17

Problem 9

Let $X_{n} \in \operatorname{Bin}\left(n, p_{n}\right)$.
(a) Suppose that $n \cdot p_{n} \rightarrow m$ as $n \rightarrow \infty$. Show that
$$
X_{n} \stackrel{d}{\longrightarrow} \operatorname{Po}(m) \quad \text { as } \quad n \rightarrow \infty .
$$
(b) Suppose that $p_{n} \rightarrow 0$ and that $n p_{n} \rightarrow \infty$ as $n \rightarrow \infty$. Show that
$$
\frac{X_{n}-n p_{n}}{\sqrt{n p_{n}}} \stackrel{d}{\longrightarrow} N(0,1) \quad \text { as } \quad n \rightarrow \infty .
$$
(c) Suppose that $n p_{n}\left(1-p_{n}\right) \rightarrow \infty$ as $n \rightarrow \infty$. Show that
$$
\frac{X_{n}-n p_{n}}{\sqrt{n p_{n}\left(1-p_{n}\right)}} \stackrel{d}{\rightarrow} N(0,1) \quad \text { as } \quad n \rightarrow \infty .
$$
Remark. These results, which usually are presented without proofs in a first probability course, verify the common approximations of the binomial distribution with the Poisson and normal distributions.

Foster Wisusik
Foster Wisusik
Numerade Educator
02:06

Problem 10

Let $X_{n} \in \operatorname{Bin}\left(n^{2}, \frac{m}{n}\right), m>0$. Show that
$$
\frac{X_{n}-n \cdot m}{\sqrt{n m}} \stackrel{d}{\longrightarrow} N(0,1) \quad \text { as } \quad n \rightarrow \infty .
$$

Bobby Barnes
Bobby Barnes
University of North Texas
02:36

Problem 11

Let $X_{n 1}, X_{n 2}, \ldots, X_{n n}$ be independent random variables, with a common distribution given as follows:
$$
P\left(X_{n k}=0\right)=1-\frac{1}{n}-\frac{1}{n^{2}}, \quad P\left(X_{n k}=1\right)=\frac{1}{n}, \quad P\left(X_{n k}=2\right)=\frac{1}{n^{2}},
$$
where $k=1,2, \ldots, n$ and $n=1,2, \ldots$ Set $S_{n}=X_{n 1}+X_{n 2}+\ldots+$ $X_{n n}, n \geq 1$. Show that
$$
S_{n} \stackrel{d}{\longrightarrow} \operatorname{Po}(1) \quad \text { as } \quad n \rightarrow \infty .
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
16:48

Problem 12

Let $X_{1}, X_{2}, \ldots$ be independent, equidistributed random variables with characteristic function
$$
\varphi(t)=\left\{\begin{array}{lll}
1-\sqrt{|t|(2-|t|)}, & \text { for } & |t| \leq 1 \\
0, & \text { for } & |t| \geq 1
\end{array}\right.
$$
Set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1$. Show that $S_{n} / n^{2}$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:48

Problem 13

Let $X_{1}, X_{2}, \ldots$ be independent, $L(a)$-distributed random variables, and let $N \in \operatorname{Po}(m)$ be independent of $X_{1}, X_{2}, \ldots$ Determine the limit distribution of $S_{N}=X_{1}+X_{2}+\ldots+X_{N}$ (where $S_{0}=0$ ) as $m \rightarrow \infty$ and $a \rightarrow 0$ in such a way that $m \cdot a^{2} \rightarrow 1$.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:32

Problem 14

Let $N, X_{1}, X_{2}, \ldots$ be independent random variables such that $N \in$ $\operatorname{Po}(\lambda)$ and $X_{k} \in \operatorname{Po}(\mu), k=1,2, \ldots$ Determine the limit distribution of $X_{1}+X_{2}+\ldots+X_{N}$ as $\lambda \rightarrow \infty$ and $\mu \rightarrow 0$ such that $\lambda \cdot \mu \rightarrow \gamma>0$. (The sum equals 0 for $N=0$.)

Manik Pulyani
Manik Pulyani
Numerade Educator
07:49

Problem 15

Let $X_{1}, X_{2}, \ldots$ be independent, $\operatorname{Po}(m)$-distributed random variables, and suppose that $N \in \operatorname{Ge}(p)$ is independent of $X_{1}, X_{2}, \ldots$Put $S_{N}=X_{1}+X_{2}+\ldots+X_{N}$ (and $S_{0}=0$ for $N=0$ ). Let $m \rightarrow 0$ and $p \rightarrow 0$ in such a way that $\frac{p}{m} \rightarrow \alpha$, where $\alpha$ is a given positive number. Show that $S_{N}$ converges in distribution, and determine the limit distribution.

Abhirup Pal
Abhirup Pal
Numerade Educator
16:48

Problem 16

Suppose that the random variables $N_{n}, X_{1}, X_{2}, \ldots$ are independent, that $N_{n} \in \operatorname{Ge}\left(p_{n}\right), 0<p_{n}<1$, and that $X_{1}, X_{2}, \ldots$ are identically distributed with finite mean $\mu$. Show that if $p_{n} \rightarrow 0$ as $n \rightarrow \infty$ then $p_{n}\left(X_{1}+X_{2}+\ldots X_{N_{n}}\right)$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Mengchun Cai
Mengchun Cai
Numerade Educator
16:48

Problem 17

Let $X_{1}, X_{2}, \ldots$ be independent, $U(0,1)$-distributed random variables, and let $N_{m} \in \operatorname{Po}(m)$ be independent of $X_{1}, X_{2}, \ldots$ Set
$$
V_{m}=\max \left\{X_{1}, \ldots, X_{N_{m}}\right\}
$$
$\left(V_{m}=0\right.$ when $\left.N_{m}=0\right)$. Determine
(a) the distribution function of $V_{m}$,
(b) the moment generating function of $V_{m}$.
It is reasonable to believe that $V_{m}$ is "close" to 1 when $m$ is "large" (cf. Problem 2). The purpose of parts (c) and (d) is to show how this can be made more precise.
(c) Show that $E V_{m} \rightarrow 1$ as $m \rightarrow \infty$.
(d) Show that $m\left(1-V_{m}\right)$ converges in distribution as $m \rightarrow \infty$, and determine the limit distribution.

Mengchun Cai
Mengchun Cai
Numerade Educator
02:36

Problem 18

Let $X_{1 n}, X_{2 n}, \ldots, X_{n n}$ be independent random variables such that $X_{k n} \in \operatorname{Be}\left(p_{k, n}\right), k=1,2, \ldots, n, n \geq 1$. Suppose, further, that $\sum_{k=1}^{n} p_{k, n} \rightarrow \lambda<\infty$ and that $\max _{1 \leq k \leq n} p_{k, n} \rightarrow 0$ as $n \rightarrow \infty$. Show that $\sum_{k=1}^{n} X_{k n}$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:05

Problem 19

In a game of dice one wishes to use one of two dice, $A$ and $B . A$ has two white and four red faces and $B$ has two red and four white faces. A coin is tossed in order to decide which die is to be used and that die is then used throughout. Let $\left\{X_{k}, k \geq 1\right\}$ be a sequence of random variables defined as follows:
$$
X_{k}= \begin{cases}1 & \text { if red is obtained, } \\ 0 & \text { if white is obtained }\end{cases}
$$
at the $k$ th roll of the die. Show that the law of large numbers does not hold for the sequence $\left\{X_{k}, k \geq 1\right\}$. Why is this the case?

Lucas Finney
Lucas Finney
Numerade Educator
01:47

Problem 20

Let $X_{1}, X_{2}, \ldots$ be independent, equidistributed random variables, and set $S_{n}=X_{1}+\cdots+X_{n}, n \geq 1$. The sequence $\left\{S_{n}, n \geq 0\right\}$ (where $S_{0}=0$ ) is called a random walk. Consider the following "perturbed" random walk. Let $\left\{\varepsilon_{n}, n \geq 1\right\}$ be a sequence of random variables such that, for some fixed $A>0$, we have $P\left(\left|\varepsilon_{n}\right| \leq A\right)=1$ for all $n$, and set
$$
T_{n} \equiv S_{n}+\varepsilon_{n}, \quad n=1,2, \ldots
$$
Suppose that $E X_{1}=\mu$ exists. Show that the law of large numbers holds for the perturbed random walk, $\left\{T_{n}, n \geq 1\right\}$.

James Kiss
James Kiss
Numerade Educator
02:36

Problem 21

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed random variables with finite mean $\mu$, and let $\left\{\left(a_{n k}, 1 \leq k \leq n\right), n \geq 1\right\}$ be "weights," that is, suppose that $a_{n k} \geq 0$ and $\sum_{k=1}^{n} a_{n k}=1$, for $n=1,2, \ldots$ Suppose, in addition, that $n \cdot \max _{1 \leq k \leq n} a_{n k} \leq C$, for all $n$ (for some positive constant $C$ ), and set
$$
S_{n}=\sum_{k=1}^{n} a_{n k} X_{k}, \quad n=1,2, \ldots
$$
Prove the so-called law of large numbers for weighted sums, that is, show that

Amany Waheeb
Amany Waheeb
Numerade Educator
04:41

Problem 22

Show that
$$
\lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2}
$$
by applying the central limit theorem to suitably chosen, independent, Poisson distributed random variables.

Mengchun Cai
Mengchun Cai
Numerade Educator
02:36

Problem 23

Let $X_{1}, X_{2}, \ldots$ be independent, $U(-1,1)$-distributed random variables.
(a) Show that
$$
Y_{n}=\frac{\sum_{k=1}^{n} X_{k}}{\sum_{k=1}^{n} X_{k}^{2}+\sum_{k=1}^{n} X_{k}^{3}}
$$
converges in probability as $n \rightarrow \infty$, and determine the limit.
(b) Show that $Y_{n}$, suitably normalized, converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:05

Problem 24

Let $X_{n} \in \Gamma(n, 1)$, and set
$$
Y_{n}=\frac{X_{n}-n}{\sqrt{X_{n}}}
$$
Show that $Y_{n} \stackrel{d}{\longrightarrow} N(0,1)$ as $n \rightarrow \infty .$

Vishnu P
Vishnu P
Numerade Educator
02:36

Problem 25

Let $\left\{Y_{k}, k \geq 1\right\}$ be independent, $U(-1,1)$-distributed random variables, and set
$$
X_{n}=\frac{\sum_{1}^{n} Y_{k}}{\sqrt{n} \cdot \max _{1 \leq k \leq n} Y_{k}} .
$$
Show that $X_{n} \stackrel{d}{\longrightarrow} N\left(0, \frac{1}{3}\right)$ as $n \rightarrow \infty$.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:36

Problem 26

Let $X_{1}, X_{2}, \ldots$ be independent, $U(-a, a)$-distributed random variables $(a>0)$. Set
$$
S_{n}=\sum_{k=1}^{n} X_{k}, \quad Z_{n}=\max _{1 \leq k \leq n} X_{k}, \quad \text { and } \quad V_{n}=\min _{1 \leq k \leq n} X_{k} .
$$
Show that $S_{n} Z_{n} / V_{n}$, suitably normalized, converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:07

Problem 27

Let $X_{1}, X_{2}, \ldots$ be independent, $U(0,1)$-distributed random variables, and set
$$
Z_{n}=\max _{1 \leq k \leq n} X_{k} \text { and } V_{n}=\min _{1 \leq k \leq n} X_{k}
$$
Determine the limit distribution of $n V_{n} / Z_{n}$ as $n \rightarrow \infty$.

James Kiss
James Kiss
Numerade Educator
02:36

Problem 27

Let $X_{1}, X_{2}, \ldots$ be independent random variables such that $X_{k} \in$ $\operatorname{Exp}(k !), k=1,2 \ldots$, and set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1$. Show that
$$
\frac{S_{n}}{n !} \stackrel{d}{\longrightarrow} \operatorname{Exp}(1) \quad \text { as } \quad n \rightarrow \infty .
$$
Hint. What is the distribution of $X_{n} / n !$ ?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:36

Problem 28

Let $X_{1}, X_{2}, \ldots$ be independent random variables such that $X_{k} \in$ $\operatorname{Exp}(k !), k=1,2 \ldots$, and set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1$. Show that
$$
\frac{S_{n}}{n !} \stackrel{d}{\longrightarrow} \operatorname{Exp}(1) \quad \text { as } \quad n \rightarrow \infty .
$$
Hint. What is the distribution of $X_{n} / n !$ ?

Amany Waheeb
Amany Waheeb
Numerade Educator
16:48

Problem 29

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed random variables with expectation 1 and finite variance $\sigma^{2}$, and set $S_{n}=X_{1}+$ $X_{2}+\ldots+X_{n}$, for $n \geq 1 .$ Show that
$$
\sqrt{S_{n}}-\sqrt{n} \stackrel{d}{\longrightarrow} N\left(0, b^{2}\right) \quad \text { as } \quad n \rightarrow \infty,
$$
and determine the constant $b^{2}$.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:48

Problem 30

Let $X_{1}, X_{2}, \ldots$ be independent, equidistributed random variables with mean $\mu$ and variance $\sigma^{2}$, both finite. The relation
$$
\sum_{k=1}^{n}\left(X_{k}-\mu\right)^{2}=\sum_{k=1}^{n}\left(X_{k}-\bar{X}_{n}\right)^{2}+n\left(\bar{X}_{n}-\mu\right)^{2},
$$
where $\bar{X}_{n}=\left(X_{1}+X_{2}+\ldots X_{n}\right) / n$, is useful in various situations (cf. Section V.9). Put $s_{n}^{2}=\frac{1}{n-1} \sum_{k=1}^{n}\left(X_{k}-\bar{X}_{n}\right)^{2}$.
(a) Show that
$$
s_{n}^{2} \stackrel{p}{\longrightarrow} \sigma^{2} \quad \text { as } \quad n \rightarrow \infty .
$$
(b) Suppose, in addition, that $\mu_{4}=E\left(X_{1}-\mu\right)^{4}<\infty$. Show that
$$
\sqrt{n}\left(s_{n}^{2}-\sigma^{2}\right) \stackrel{d}{\longrightarrow} N\left(0, \mu_{4}-\sigma^{4}\right) \quad \text { as } \quad n \rightarrow \infty .
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:27

Problem 31

Suppose that $N \in \operatorname{Po}(\lambda)$ independent observations on a random variable, $X$, with mean 0 and variance 1 are performed. Moreover, assume that $N$ is independent of $X_{1}, X_{2}, \ldots$ Show that
$$
\frac{X_{1}+X_{2}+\ldots+X_{N}}{\sqrt{N}} \stackrel{d}{\longrightarrow} N(0,1) \text { as } \lambda \rightarrow \infty .
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
03:06

Problem 32

Let $X_{1}, X_{2}, \ldots$ be independent random variables such that, for some fixed positive integer, $m, X_{1}, \ldots, X_{m}$ are equidistributed with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$ and $X_{m+1}, X_{m+2}, \ldots$ are equidistributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2} .$ Set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1 .$ Show that the central limit theorem (still) holds.
Remark. Begin with the case $m=1$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:28

Problem 33

Let $X_{1}, X_{2}, \ldots$ be $U(-1,1)$-distributed random variables, and set
$$
Y_{n}= \begin{cases}X_{n}, & \text { for }\left|X_{n}\right| \leq 1-\frac{1}{n} \\ n & \text { otherwise }\end{cases}
$$
(a) Show that $Y_{n}$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.
(b) Let $Y$ denote the limiting random variable. Consider the statements $E Y_{n} \rightarrow E Y$ and $\operatorname{Var} Y_{n} \rightarrow \operatorname{Var} Y$ as $n \rightarrow \infty$. Are they true or false?

Nick Johnson
Nick Johnson
Numerade Educator
07:49

Problem 34

Let $X_{1}, X_{2}, \ldots$ be independent random variables such that $X_{k} \in$ $\operatorname{Po}(k), k=1,2, \ldots$, and set $Y_{n}=\frac{1}{n}\left\{\sum_{k=1}^{n} X_{k}-\frac{n^{2}}{2}\right\} .$ Show that $Y_{n}$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Abhirup Pal
Abhirup Pal
Numerade Educator
03:24

Problem 35

The random variable $X$ has zero mean and all moments exist. The sequence $\left\{X_{k}, k \geq 1\right\}$ of independent random variables has the property that
$$
X_{k} \stackrel{d}{=} \sqrt{k} \cdot X .
$$
Show that $\sum_{k=1}^{n} X_{k}$ is asymptotically normal as $n \rightarrow \infty$.

Shreya Kelly
Shreya Kelly
Numerade Educator
11:52

Problem 36

The purpose of this problem is to show that one can obtain a central limit theorem even if the summands have infinite variance (if the variance does not exist). Namely, let $X_{1}, X_{2}, \ldots$ be independent random variables with the following symmetric Pareto distribution:
$$
f_{X}(x)= \begin{cases}\frac{1}{|x|^{3}}, & \text { for }|x|>1 \\ 0 & \text { otherwise }\end{cases}
$$
Set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1$. Show via the following steps that
$$
\frac{S_{n}}{\sqrt{n \log n}} \stackrel{d}{\rightarrow} N(0,1) \quad \text { as } \quad n \rightarrow \infty .
$$
(Note that we do not normalize by $\sqrt{n}$ as in the standard case.)
Fix $n$ and consider, for $k=1,2, \ldots, n$, the truncated random variables
$$
Y_{n k}= \begin{cases}X_{k} & \text { when }\left|X_{k}\right| \leq \sqrt{n} \\ 0 & \text { otherwise }\end{cases}
$$
and
$$
Z_{n k}= \begin{cases}X_{k} & \text { when }\left|X_{k}\right|>\sqrt{n} \\ 0 & \text { otherwise }\end{cases}
$$
(and note that $Y_{n k}+Z_{n k}=X_{k}$ ). Further, set $S_{n}^{\prime}=\sum_{k=1}^{n} Y_{n k}$ and $S_{n}^{\prime \prime}=\sum_{k=1}^{n} Z_{n k} .$
(a) Show that
$$
E\left|\frac{S_{n}^{\prime \prime}}{\sqrt{n \log n}}\right| \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty
$$
and conclude that
$$
\frac{S_{n}^{\prime \prime}}{\sqrt{n \log n}} \rightarrow 0 \text { in 1-mean as } n \rightarrow \infty
$$
and hence in probability (why?).
(b) Show that it remains to prove that
$$
\frac{S_{n}^{\prime}}{\sqrt{n \log n}} \stackrel{d}{\longrightarrow} N(0,1) \text { as } n \rightarrow \infty .
$$
(c) Let $\varphi$ denote a characteristic function. Show that
$$
\varphi_{Y_{n k}}(t)=1-2 \int_{1}^{\sqrt{n}} \frac{1-\cos t x}{x^{3}} d x
$$
and hence that
$$
\varphi \frac{s_{\vdots}^{\prime}}{\sqrt{n \log n}}(t)=\left(1-2 \int_{1}^{\sqrt{n}} \frac{1-\cos \frac{t x}{\sqrt{n \log n}}}{x^{3}} d x\right)^{n} .
$$
(d) Show that it remains to prove
$$
2 \int_{1}^{\sqrt{n}} \frac{1-\cos \frac{t x}{\sqrt{n \log n}}}{x^{3}} d x=\frac{t^{2}}{2 n}+o\left(\frac{1}{n}\right) \quad \text { as } \quad n \rightarrow \infty .
$$
(e) Prove this relation.
Remark. Note that (a) and (b) together show that $S_{n}$ and $S_{n}^{\prime}$ have the same asymptotic distributional behavior, that $\operatorname{Var} Y_{n k}=\log n$ for $1 \leq k \leq n$ and $n \geq 1$, and hence that we have used the "natural" normalization $\sqrt{\operatorname{Var} S_{n}^{\prime}}$ for $S_{n}^{\prime}$.

Victor Salazar
Victor Salazar
Numerade Educator
01:48

Problem 37

The "usual" central limit theorem is concerned with normed sums of independent, identically distributed random variables. Prove the following central limit theorem for a sum of independent (not identically distributed) random variables.
Let $X_{1}, X_{2}, \ldots$ be independent random variables such that $X_{k} \in$ $U(-k, k)$, and set $S_{n}=\sum_{k=1}^{n} X_{k}, n \geq 1 .$ Show that
$$
\frac{S_{n}}{n^{\frac{3}{2}}} \stackrel{d}{\longrightarrow} N\left(\mu, \sigma^{2}\right) \quad \text { as } \quad n \rightarrow \infty
$$
and determine $\mu$ and $\sigma^{2}$.
Remark 1. It may be useful to recall that $\sin x=x-x^{3} / 3 !+R(x)$, where $|R(x)| \leq|x|^{5} / 5 !$, and that $1^{2}+2^{2}+\ldots+n^{2}=n(n+1)(2 n+1) / 6$ Remark 2. Note that the normalization is not proportional to $\sqrt{n}$; rather, it is asymptotically proportional to $\sqrt{\operatorname{Var} S_{n}}$.

Manik Pulyani
Manik Pulyani
Numerade Educator