Math Problems Vault

Problem 5.1

(a)(b)(c)

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(a)

Show that $X$ is measurable with respect to the $\sigma$ -field $\mathscr {A}$ if and only if $\sigma (X) \subset \mathscr {G}$ . Show that $X$ is measurable $\sigma (Y)$ if and only if $\sigma (X) \subset \sigma (Y)$ .

(b)

Show that, if $\mathscr {G}=\left\{ \varnothing , \Omega \right\}$ , then $X$ is measurable $\mathscr {G}$ if and only if $X$ is constant.

(c)

Suppose that $\mathbb {P}\left(A\right)$ is 0 or 1 for every $A$ in $\mathscr {G}$ . This holds, for example, if $\mathscr {G}$ is the tail field of an independent sequence (Theorem 4.5), or if $\mathscr {E}$ consists of the countable and cocountable sets on the unit interval with Lebesgue measure. Show that if $X$ is measurable $\mathscr {G}$ , then $\mathbb {P}\left(X=c\right)=1$ for some constant $c$ .

Problem 5.2

$2.19 \uparrow$ Show that the unit interval can be replaced by any nonatomic probability measure space in the proof of Theorem 5.3.

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Problem 5.3

Show that $m=\mathbb {E}\left[X\right]$ minimizes $\mathbb {E}\left[(X-m)^{2}\right]$ .

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Problem 5.4

Suppose that $X$ assumes the values $m-\alpha , m, m+\alpha$ with probabilities $p, 1- 2 p, p$ , and show that there is equality in (5.32). Thus Chebyshev's inequality cannot be improved without special assumptions on $X$ .

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Problem 5.5

(a)(b)(c)

Suppose that $X$ has mean $m$ and variance $\sigma^{2}$ .

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(a)

Prove Cantelli's inequality

\mathbb {P}\left(X-m \geq \alpha \right) \leq \frac{\sigma ^{2}}{\sigma ^{2}+\alpha ^{2}}, \quad \alpha \geq 0.

(b)

Show that $\mathbb {P}\left(\left|X-m\right| \geq \alpha \right) \leq 2 \sigma^{2} /\left(\sigma^{2}+\alpha^{2}\right)$ . When is this better than Chebyshev's inequality?

(c)

By considering a random variable assuming two values, show that Cantelli's inequality is sharp.

Problem 5.6

The polynomial $\mathbb {E}\left[(t\left|X\right|+\left|Y\right|)^{2}\right]$ in $t$ has at most one real zero. Deduce Schwarz's inequality once more.

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Problem 5.7

(a)(b)

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(a)

Write (5.37) in the form $\left.E^{\beta / \alpha }\left[|X|^{\alpha }\right] \leq E\left[|X|^{\alpha }\right)^{\beta / \alpha }\right]$ and deduce it directly from Jensen's inequality.

(b)

Prove that $\mathbb {E}\left[1 / X^{p}\right] \geq 1 / E^{p}[X]$ for $p>0$ and $X$ a positive random variable.

Problem 5.8

(a)(b)

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(a)

Let $f$ be a convex real function on a convex set $C$ in the plane Suppose that $(X(\omega ), Y(\omega )) \in C$ for all $\omega$ and prove a two-dimensional Jensen's mequality:

f(\mathbb {E}\left[X\right], \mathbb {E}\left[Y\right]) \leq \mathbb {E}\left[f(X, Y)\right]. \tag {5.38}

(b)

Show that $f$ is convex if it has continuous second derivatives that satisfy

f_{11} \geq 0, \quad f_{22} \geq 0, \quad f_{11} f_{22} \geq f_{12}^{2}. \tag {5.39}

Problem 5.9

$\uparrow$ Hölder's inequality is equivalent to $\mathbb {E}\left[X^{1 / p} Y^{1 / q}\right] \leq E^{1 / p}[X] \cdot E^{1 / q}[Y] \left(p^{-1}+q^{-1}=1\right)$ , where $X$ and $Y$ are nonnegative random variables. Derive this from (5.38).

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Problem 5.10

↑ Minkowski's inequality is

E^{1 / p}\left[|X+Y|^{p}\right] \leq E^{1 / p}\left[|X|^{p}\right]+E^{1 / p}\left[|Y|^{p}\right], \tag {5.40}

valid for $p \geq 1$ . It is enough to prove that $\mathbb {E}\left[\left(X^{1 / p}+Y^{1 / p}\right)^{p}\right] \leq \left(E^{1 / p}[X]+\right. \left.E^{1 / p}[Y]\right)^{p}$ for nonnegative $X$ and $Y$ . Use (5.38).

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Problem 5.11

For events $A_{1}, A_{2}, \ldots$ , not necessarily independent, let $N_{n}=\sum_{k=1}^{n} i_{A_{k}}$ be the number to occur among the first $n$ . Let

\alpha _{n}=\frac{1}{n} \sum _{k=1}^{n} \mathbb {P}\left(A_{k}\right), \quad \beta _{n}=\frac{2}{n(n-1)} \sum _{1 \leq j<k \leq n} \mathbb {P}\left(A_{j} \cap A_{k}\right). \tag {5.41}

Show that

\mathbb {E}\left[n^{-1} N_{n}\right]=\alpha _{n}, \quad \operatorname {Var}\left[n^{-1} N_{n}\right]=\beta _{n}-\alpha _{n}^{2}+\frac{\alpha _{n}-\beta _{n}}{n}. \tag {5.42}

Thus $\operatorname {Var}\left[n^{-1} N_{n}\right] \rightarrow 0$ if and only if $\beta_{n}-\alpha_{n}^{2} \rightarrow 0$ , which holds if the $A_{n}$ are independent and $\mathbb {P}\left(A_{n}\right)=p$ (Bernoulli trials), because then $\alpha_{n}=p$ and $\beta_{n}= p^{2}=\alpha_{n}^{2}$ .

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Problem 5.12

Show that, if $X$ has nonnegative integers as values, then $\mathbb {E}\left[X\right]=\sum_{n=1}^{\infty } \mathbb {P}\left(X \geq n\right)$ .

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Problem 5.13

Let $I_{i}=I_{A_{i}}$ be the indicators of $n$ events having union $A$ . Let $S_{k}=\sum I_{i_{1}} \cdots I_{i_{k}}$ , where the summation extends over all $k$ -tuples satisfying $1 \leq i_{1}<\cdots <i_{k} \leq n$ . Then $s_{k}=\mathbb {E}\left[S_{k}\right]$ are the terms in the inclusion-exclusion formula $\mathbb {P}\left(A\right)=s_{1}- s_{2}+\cdots \pm s_{n}$ . Deduce the inclusion-exclusion formula from $I_{A}=S_{1}-S_{2}+ \cdots \pm S_{n}$ . Prove the latter formula by expanding the product $\prod_{i=1}^{n}\left(1-I_{i}\right)$

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Problem 5.14

Let $f_{n}(x)$ be $n^{2} x$ or $2 n-n^{2} x$ or 0 according as $0 \leq x \leq n^{-1}$ or $n^{-1} \leq x \leq 2 n^{-1}$ or $2 n^{-1} \leq x \leq 1$ . This gives a standard example of a sequence of continuous functions that converges to 0 but not uniformly. Note that $\int_{0}^{1} f_{n}(x) d x$ does not converge to 0 ; relate to Example 5.7.

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Problem 5.15

By Theorem 5.3, for any prescribed sequence of probabilities $p_{n}$ , there exists (on some space) an independent sequence of events $A_{n}$ satisfying $\mathbb {P}\left(A_{n}\right)=p_{n}$ . Show that if $p_{n} \rightarrow 0$ but $\sum p_{n}=\infty$ , this gives a counterexample (like Example 5.4) to the converse of Theorem 5.2(ii).

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Problem 5.16

$\uparrow$ Suppose that $0 \leq p_{n} \leq 1$ and put $\alpha_{n}=\min \left\{ p_{n}, 1-p_{n}\right\}$ . Show that, if $\sum \alpha_{n}$ converges, then on some discrete probability space there exist independent events $A_{n}$ satisfying $\mathbb {P}\left(A_{n}\right)=p_{n}$ . Compare Problem 1.1(b).

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Problem 5.17

(a)(b)

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(a)

Suppose that $X_{n} \rightarrow_{P} X$ and that $f$ is continuous. Show that $f\left(X_{n}\right) \rightarrow_{P} f(X)$ .

(b)

Show that $\mathbb {E}\left[\left|X-X_{n}\right|\right] \rightarrow 0$ implies $X_{n} \rightarrow_{p} X$ . Show that the converse is false.

Problem 5.18

$2.20 \uparrow$ The proof given for Theorem 5.3 for the special case where the $\mu_{n}$ are all the same can be extended to cover the general case: use Problem 2.20.

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Problem 5.19

$2.18 \uparrow$ For integers $m$ and primes $p$ , let $\alpha_{p}(m)$ be the exact power of $p$ in the prime factorization of $m: m=\Pi_{p} p^{\alpha_{p}(m)}$ . Let $\delta_{p}(m)$ be 1 or 0 as $p$ divides $m$ or not. Under each $P_{n}$ (see (2.34)) the $\alpha_{p}$ and $\delta_{p}$ are random variables. Show that for distinct primes $p_{1}, \ldots , p_{u}$ ,

\mathbb {P}_{n}\left(\alpha _{p_{i}} \geq k_{i}, i \leq u\right)=\frac{1}{n}\left\lfloor \frac{n}{p_{1}^{k_{1}} \cdots p_{u}^{k_{n}}}\right\rfloor \rightarrow \frac{1}{p_{1}^{k_{1}} \cdots p_{u}^{k_{n}}} \tag {5.43}

and

\mathbb {P}_{n}\left(\alpha _{p_{i}}=k_{i}, i \leq u\right) \rightarrow \prod _{i=1}^{u}\left(\frac{1}{p_{i}^{k_{i}}}-\frac{1}{p_{i}^{k_{i}+1}}\right). \tag {5.44}

Similarly,

\mathbb {P}_{n}\left(\delta _{p_{1}}=1, i \leq u\right)=\frac{1}{n}\left|\frac{n}{p_{1} \cdots p_{u}}\right| \rightarrow \frac{1}{p_{1} \cdots p_{u}} \tag {5.45}

According to (5.44), the $\alpha_{p}$ are for large $n$ approximately independent under $P_{n}$ , and according to (5.45), the same is true of the $\delta_{p}$ .

For a function $f$ of positive integers, let

E_{n}[f]=\frac{1}{n} \sum _{m=1}^{n} f(m) \tag {5.46}

be its expected value under the probability measure $P_{n}$ . Show that

E_{n}\left[\alpha _{p}\right]=\sum _{k=1}^{\infty } \frac{1}{n}\left|\frac{n}{p^{k}}\right| \rightarrow \frac{1}{p-1} ; \tag {5.47}

this says roughly that $(p-1)^{-1}$ is the average power of $p$ in the factorization of large integers.

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Problem 5.20

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(a)

From Stirling's formula, deduce

E_{n}[\log ]=\log n+O(1). \tag {5.48}

From this, the inequality $E_{n}\left[\alpha_{p}\right] \leq 2 / p$ , and the relation $\log m=\sum_{p} \alpha_{p}(m) \log p$ , conclude that $\Sigma_{p} p^{-1} \log p$ diverges and that there are infinitely many primes.

(b)

Let $\log^{*} m=\sum_{p} \delta_{p}(m) \log p$ . Show that

E_{n}\left[\log ^{*}\right]=\sum _{p} \frac{1}{n}\left\lfloor \frac{n}{p}\right\rfloor \log p=\log n+O(1). \tag {5.49}

(c)

Show that $\lfloor 2 n / p\rfloor -2\lfloor n / p\rfloor$ is always nonnegative and equals 1 in the range $n<p \leq 2 n$ . Deduce $E_{2 n}\left[\log^{*}\right]-E_{n}\left[\log^{*}\right]=O(1)$ and conclude that

\sum _{p \leq x} \log p=O(x). \tag {5.50}

Use this to estimate the error removing the integral-part brackets introduces into (5.49), and show that

\sum _{p \leq x} p^{-1} \log p=\log x+O(1). \tag {5.51}

(d)

Restrict the range of summation in (5.51) to $\theta x<p \leq x$ for an appropriate $\theta$ , and conclude that

\sum _{p \leq x} \log p \approx x, \tag {5.52}

in the sense that the ratio of the two sides is bounded away from 0 and $\infty$ .

(e)

Use (5.52) and truncation arguments to prove for the number $\pi (x)$ of primes not exceeding $x$ that

\pi (x) \asymp \frac{x}{\log x}. \tag {5.53}

(By the prime number theorem the ratio of the two sides in fact goes to 1.) Conclude that the $r$ th prime $p_{r}$ satisfies $p_{r} \asymp r \log r$ and that

\sum _{p} \frac{1}{p}=\infty . \tag {5.54}