Math Problems Vault

Problem 14.1

The general nondecreasing function $F$ has at most countably many discontinuities. Prove this by considering the open intervals

\left(\sup _{u<x} F(u), \inf _{\iota >x} F(v)\right)

-each nonempty one contains a rational.

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Problem 14.2

(a)(b)

For distribution functions $F$ , the second proof of Theorem 14.1 shows how to construct a measure $\mu$ on $(R^{1}, \mathscr {R}^{1})$ such that $\mu (a, b]=F(b)-F(a)$ .

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(a)

Extend to the case of bounded $F$ .

(b)

Extend to the general case. Hint: Let $F_{n}(x)$ be $-n$ or $F(x)$ or $n$ as $F(x)<-n$ or $-n \leq F(x)<n$ or $n \leq F(x)$ . Construct the corresponding $\mu_{n}$ and define $\mu (A)=\lim_{n} \mu_{n}(A)$ .

Problem 14.3

(a)(b)(c)

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(a)

Suppose that $X$ has a continuous, strictly increasing distribution function $F$ . Show that the random variable $F(X)$ is uniformly distributed over the unit interval in the sense that $\mathbb {P}\left(F(X) \leq u\right)=u$ for $0 \leq u \leq 1$ . Passing from $X$ to $F(X)$ is called the probability transformation.

(b)

Show that the function $\varphi (u)$ defined by (14.5) satisfies $F(\varphi (u)-) \leq u \leq F(\varphi (u))$ and that, if $F$ is continuous (but not necessarily strictly increasing), then $F(\varphi (u))=u$ for $0<u<1$ .

(c)

Show that $\mathbb {P}\left(F(X)<u\right)=F(\varphi (u)-)$ and hence that the result in part (a) holds as long as $F$ is continuous.

${ }^{\dagger }$ This theory is associated with the names of Fisher, Fréchet, Gnedenko, and Tippet. For further information, see Galambos.

Problem 14.4

(a)(b)

↑ Let $C$ be the set of continuity points of $F$ .

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(a)

Show that for every Borel set $A, \mathbb {P}\left(F(X) \in A, X \in C\right)$ is at most the Lebesgue measure of $A$ .

(b)

Show that if $F$ is continuous at each point of $F^{-1} A$ , then $\mathbb {P}\left(F(X) \in A\right)$ is at most the Lebesgue measure of $A$ .

Problem 14.5

The Lévy distance $d(F, G)$ between two distribution functions is the infimum of those $\epsilon$ such that $G(x-\epsilon )-\epsilon \leq F(x) \leq G(x+\epsilon )+\epsilon$ for all $x$ . Verify that this is a metric on the set of distribution functions. Show that a necessary and sufficient condition for $F_{n} \Rightarrow F$ is that $d\left(F_{n}, F\right) \rightarrow 0$ .

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Problem 14.6

12.3 ↑ A Borel function satisfying Cauchy's equation [A20] is automatically bounded in some interval and hence satisfies $f(x)=x f(1)$ . Hint: Take $K$ large enough that $\lambda [x: x>s,\left|f(x)\right| \leq K]>0$ . Apply Problem 12.3 and conclude that $f$ is bounded in some interval to the right of 0

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Problem 14.7

(a)(b)(c)

↑ Consider sets $S$ of reals that are !inearly independent over the field of rationals in the sense that $n_{1} x_{1}+\cdots +n_{k} x_{k}=0$ for distinct points $x_{i}$ in $S$ and integers $n_{i}$ (positive or negative) is impossible unless $n_{i} \equiv 0$ .

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(a)

By Zorn's lemma find a maximal such $S$ . Show that it is a Hamel basis. That is, show that each real $x$ can be written uniquely as $x=n_{1} x_{1}+\cdots +n_{k} x_{k}$ for distinct points $x_{i}$ in $S$ and integers $n_{i}$ .

(b)

Define $f$ arbitrarily on $S$ , and define it elsewhere by $f\left(n_{1} x_{1}+\cdots +n_{k} x_{k}\right) =n_{1} f\left(x_{1}\right)+\cdots +n_{k} f\left(x_{k}\right)$ . Show that $f$ satisfies Cauchy's equation but need not satisfy $f(x)=x f(1)$ .

(c)

By means of Problem 14.6 give a new construction of a nonmeasurable function and a nonmeasurable set.

Problem 14.8

(a)(b)(c)

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(a)

$14.5 \uparrow$ Show that if a distribution function $F$ is everywhere continuous, then it is uniformly continuous.

(b)

Let $\delta_{F}(\epsilon )=\sup [F(x)-F(y):\left|x-y\right| \leq \epsilon ]$ be the modulus of continuity of $F$ . Show that $d(F, G)<\epsilon$ implies that $\sup_{x}\left|F(x)-G(x)\right| \leq \epsilon +\delta_{F}(\epsilon )$ .

(c)

Show that, if $F_{n} \Rightarrow F$ and $F$ is everywhere continuous, then $F_{n}(x) \rightarrow F(x)$ uniformly in $x$ . What if $F$ is continuous over a closed interval?

Problem 14.9

Show that (14.24) and (14.25) are everywhere infinitely differentiable, although not analytic.

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