Math Problems Vault

Problem 25.1

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(a)

Show by example that distribution functions having densities can converge weakly even if the densities do not converge: Hint: Consider $f_{n}(x)=1+ \cos 2 \pi n x$ on $[0,1]$ .

(b)

Let $f_{n}$ be $2^{n}$ times the indicator of the set of $x$ in the unit interval for which $d_{n+1}(x)=\cdots =d_{2 n}(x)=0$ , where $d_{k}(x)$ is the $k$ th dyadic digit. Show that $f_{n}(x) \rightarrow 0$ except on a set of Lebesgue measure 0 ; on this exceptional set, redefine $f_{n}(x)=0$ for all $n$ , so that $f_{n}(x) \rightarrow 0$ everywhere. Show that the distributions corresponding to these densities converge weakly to Lebesgue measure confined to the unit interval.

(c)

Show that distributions with densities can converge weakly to a limit that has no density (even to a unit mass).

(d)

Show that discrete distributions can converge weakly to a distribution that has a density.

(e)

Construct an example, like that of Example 25.3, in which $\mu_{n}(A) \rightarrow \mu (A)$ fails but in which all the measures come from continuous densities on $[0,1]$ .

Problem 25.2

$14.8 \uparrow$ Give a simple proof of the Gilvenko-Cantelli theorem (Theorem 20.6) under the extra hypothesis that $F$ is continuous.

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Problem 25.3

(a)(b)(c)

Initial digits.

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(a)

Show that the first significant digit of a positive number $x$ is $d$ (in the scale of 10) if and only if $\left\{ \log_{10} x\right\}$ lies between $\log_{10} d$ and $\log_{10}(d+1)$ , $d=1, \ldots , 9$ , where the braces denote fractional part.

(b)

For positive numbers $x_{1}, x_{2}, \ldots$ , let $N_{r}(d)$ be the number among the first $n$ that have initial digit $d$ . Show that

\lim _{n} \frac{1}{n} N_{n}(d)=\log _{10}(d+1)-\log _{10} d, \quad d=1, \ldots , 9 \tag {25.14}

if the sequence $\log_{10} x_{n}, n=1,2, \ldots$ , is uniformly distributed modulo 1. This is true, for example, of $x_{n}=\vartheta^{n}$ if $\log_{10} \vartheta$ is irrational.

(c)

Let $D_{n}$ be the first significant digit of a positive random variable $X_{n}$ . Show that

\lim _{n} \mathbb {P}\left(D_{n}=d\right)=\log _{10}(d+1)-\log _{10} d, \quad d=1, \ldots , 9 \tag {25.15}

if $\left\{ \log_{10} X_{n}\right\} \Rightarrow U$ , where $U$ is uniformly distributed over the unit interval.

Problem 25.4

Show that for each probability measure $\mu$ on the line there exist probability measures $\mu_{n}$ with finite support such that $\mu_{n} \Rightarrow \mu$ . Show further that $\mu_{n}\left\{ x\right\}$ can be taken rational and that each point in the support can be taken rational. Thus there exists a countable set of probability measures such that every $\mu$ is the weak limit of some sequence from the set. The space of distribution functions is thus separable in the Lévy metric (see Problem 14.5).

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Problem 25.5

Show that (25.5) implies that $\mathbb {P}\left([X \leq x] \Delta \left[X_{n} \leq x\right]\right) \rightarrow 0$ if $\mathbb {P}\left(X=x\right)=0$ .

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Problem 25.6

For arbitrary random variables $X_{n}$ there exist positive constants $a_{n}$ such that $a_{n} X_{n} \Rightarrow 0$ .

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Problem 25.7

Generalize Example 25.8 by showing for three-dimensional random vectors $(A_{n}, B_{n}, X_{n})$ and constants $a$ and $b, a \geq 0$ , that, if $A_{n} \Rightarrow a, B_{n} \Rightarrow b$ , and $X_{n} \Rightarrow X$ , then $A_{n} X_{n}+B_{n} \Rightarrow a X+b$ Hint: First show that if $Y_{n} \Rightarrow Y$ and $D_{n} \Rightarrow 0$ , then $D_{n} Y_{n} \Rightarrow 0$ .

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Problem 25.8

Suppose that $X_{n} \Rightarrow X$ and that $h_{n}$ and $h$ are Borel functions. Let $E$ be the set of $x$ for which $h_{n} x_{n} \rightarrow h x$ fails for some sequence $x_{n} \rightarrow x$ . Suppose that $E \in \mathscr {R}^{1}$ and $\mathbb {P}\left(X \in E\right)=0$ . Show that $h_{n} X_{n} \Rightarrow h X$ .

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Problem 25.9

Suppose that the distributions of random variables $X_{n}$ and $X$ have densities $f_{n}$ and $f$ . Show that if $f_{n}(x) \rightarrow f(x)$ for $x$ outside a set of Lebesgue measure 0, then $X_{n} \Rightarrow X$ .

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Problem 25.10

$\uparrow$ Suppose that $X_{n}$ assumes as values $\gamma_{n}+k \delta_{n}, k=0, \pm 1, \ldots$ , where $\delta_{n}>0$ . Suppose that $\delta_{n} \rightarrow 0$ and that, if $k_{n}$ is an integer varying with $n$ in such a way that $\gamma_{n}+k_{n} \delta_{n} \rightarrow x$ , then $\mathbb {P}\left(X_{n}=\gamma_{n}+k_{n} \delta_{n}\right) \delta_{n}^{-1} \rightarrow f(x)$ , where $f$ is the density of a random variable $X$ . Show that $X_{n} \Rightarrow X$ .

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Problem 25.11

$\uparrow$ Let $S_{n}$ have the binomial distribution with parameters $n$ and $p$ . Assume as known that

\mathbb {P}\left(S_{n}=k_{n}\right)(n p(1-p))^{1 / 2} \rightarrow \frac{1}{\sqrt{2 \pi }} e^{-x^{2} / 2} \tag {25.16}

if $\left(k_{n}-n p\right)(n p(1-p))^{-1 / 2} \rightarrow x$ . Deduce the DeMoivre-Laplace theorem: $\left(S_{n}-n p\right)(n p(1-p))^{-1 / 2} \Rightarrow N$ , where $N$ has the standard normal distribution. This is a special case of the central limit theorem; see Section 27.

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Problem 25.12

Prove weak convergence in Example 25.3 by using Theorem 25.8 and the theory of the Riemann integral.

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Problem 25.13

(a)(b)

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(a)

Show that probability measures satisfy $\mu_{n} \Rightarrow \mu$ if $\mu_{n}(a, b] \rightarrow \mu (a, b]$ whenever $\mu \left\{ a\right\} =\mu \left\{ b\right\} =0$ .

(b)

Show that, if $\int f d \mu_{n} \rightarrow \int f d \mu$ for all continuous $f$ with bounded support, then $\mu_{n} \Rightarrow \mu$ .

Problem 25.14

$\uparrow$ Let $\mu$ be Lebesgue measure confined to the unit interval; let $\mu_{n}$ correspond to a mass of $x_{n, i}-x_{n, i-1}$ at some point in ( $x_{n, i-1}, x_{n, i}$ ], where $0=x_{n 0} <x_{n 1}<\cdots <x_{n n}=1$ . Show by considering the distribution functions that $\mu_{n} \Rightarrow \mu$ if $\max_{i<n}\left(x_{n, i}-x_{n, i-1}\right) \rightarrow 0$ . Deduce that a bounded Borel function continuous almost everywhere on the unit interval is Riemann integrable. See Problem 17.1.

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Problem 25.15

$2.18 \; 5.19 \uparrow$ A function $f$ of positive integers has distribution function $F$ if $F$ is the weak limit of the distribution function $P_{n}[m: f(m) \leq x]$ of $f$ under the measure having probability $1 / n$ at each of $1, \ldots , n$ (see 2.34)). In this case $D[m: f(m) \leq x]=F(x)$ (see (2.35)) for continuity points $x$ of $F$ . Show that $\varphi (m) / m$ (see (2.37)) has a distribution:

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(a)

Show by the mapping theorem that it suffices to prove that $f(m)= \log (\varphi (m) / m)=\sum_{p} \delta_{p}(m) \log (1-1 / p)$ has a distribution.

(b)

Let $f_{u}(m)=\sum_{p \leq u} \delta_{p}(m) \log (1-1 / p)$ , and show by (5.45) that $f_{u}$ has distribution function $F_{u}(x)=\mathbb {P}\left(\sum_{p \leq u} X_{p} \log (1-1 / p) \leq x\right)$ , where the $X_{p}$ are independent random variables (one for each prime $p$ ) such that $\mathbb {P}\left(X_{p}=1\right)= 1 / p$ and $\mathbb {P}\left(X_{\rho }=0\right)=1-1 / p$ .

(c)

Show that $\sum_{p} X_{p} \log (1-1 / p)$ converges with probability 1. Hint: Use Theorem 22.6.

(d)

Show that $\lim_{u \rightarrow \infty } \sup_{n} E_{n}\left[\left|f-f_{u}\right|\right]=0$ (see (5.46) for the notation).

(e)

Conclude by Markov's inequality and Theorem 25.5 that $f$ has the distribution of the sum in (c).

Problem 25.16

For $A \in \mathscr {R}^{1}$ and $T>0$ , put $\lambda_{T}(A)=\lambda ([-T, T] \cap A) / 2 T$ , where $\lambda$ is Lebesgue measure. The relative measure of $A$ is

\rho (A)=\lim _{T \rightarrow \infty } \lambda _{T}(A), \tag {25.17}

provided that this limit exists. This is a continuous analogue of density (see (2.35)) for sets of integers. A Borel function $f$ has a distribution under $\lambda_{T}$ ; if this converges weakly to $F$ , then

\rho [x: f(x) \leq u]=F(u) \tag {25.18}

for continuity points $u$ of $F$ , and $F$ is called the distribution function of $f$ . Show that all periodic functions have distributions.

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Problem 25.17

Suppose that $\sup_{n} \int f d \mu_{n}<\infty$ for a nonnegative $f$ such that $f(x) \rightarrow \infty$ as $x \rightarrow \pm \infty$ . Show that $\left\{ \mu_{n}\right\}$ is tight.

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Problem 25.18

$23.4 \uparrow$ Show that the random variables $A_{t}$ and $L_{t}$ in Problems 23.3 and 23.4 converge in distribution. Show that the moments converge.

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Problem 25.19

In the applications of Theorem 9.2, only a weaker result is actually needed: For each $K$ there exists a positive $\alpha =\alpha (K)$ such that if $\mathbb {E}\left[X\right]=0, \mathbb {E}\left[X^{2}\right]=1$ , and $\mathbb {E}\left[X^{4}\right] \leq K$ , then $\mathbb {P}\left(X \geq 0\right) \geq \alpha$ . Prove this by using tightness and the corollary to Theorem 25.12.

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Problem 25.20

Find uniformly integrable random variables $X_{n}$ for which there is no integrable $Z$ satisfying $\mathbb {P}\left(\left|X_{n}\right| \geq t\right) \leq \mathbb {P}\left(\left|Z\right| \geq t\right)$ for $t>0$ .

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