Math Problems Vault

Problem 22.1

Suppose that $X_{1}, X_{2}, \ldots$ is an independent sequence and $Y$ is measurable $\sigma \left(X_{n}, X_{n+1}, \ldots \right)$ for each $n$ . Show that there exists a constant $a$ such that $\mathbb {P}\left(Y=a\right)=1$ .

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Problem 22.2

Assume $\left\{ X_{n}\right\}$ independent, and define $X_{n}^{(c)}$ as in Theorem 22.8. Prove that for $\sum \left|X_{n}\right|$ to converge with probability 1 it is necessary that $\sum \mathbb {P}\left(\left|X_{n}\right|>c\right)$ and $\sum \mathbb {E}\left[\mid X_{n}^{(c)}\right] \mid$ converge for all positive $c$ and sufficient that they converge for some positive $c$ . If the three series (22.13) converge but $\Sigma \mathbb {E}\left[\left|X_{n}^{(c)}\right|\right]=\infty$ , then there is probability 1 that $\sum X_{n}$ converges conditionally but not absolutely.

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Problem 22.3

(a)(b)

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(a)

$\uparrow$ Generalize the Borel-Cantelli lemmas: Suppose $X_{n}$ are nonnegative If $\sum \mathbb {E}\left[X_{n}\right]<\infty$ , then $\sum X_{n}$ converges with probability 1. If the $X_{n}$ are indepen dent and uniformly bounded, and if $\sum \mathbb {E}\left[X_{n}\right]=\infty$ , then $\sum X_{n}$ diverges witk probability 1.

(b)

Construct independent, nonnegative $X_{n}$ such that $\sum X_{n}$ converges with probability 1 but $\sum \mathbb {E}\left[X_{n}\right]$ diverges For an extreme example, arrange that $\mathbb {P}\left(X_{n}>0 \text{ i.o.}\right)=0$ but $\mathbb {E}\left[X_{n}\right] \equiv \infty$ .

Problem 22.4

Show under the hypothesis of Theorem 22.6 that $\sum X_{n}$ has finite variance and extend Theorem 22.4 to infinite sequences.

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Problem 22.5

(a)(b)

$20.1422 .1 \uparrow$ Suppose that $X_{1}, X_{2}, \ldots$ are independent, each with the Cauchy distribution (20.45) for a common value of $u$ .

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(a)

Show that $n^{-1} \sum_{k=1}^{n} X_{k}$ does not converge with probability 1. Contrast with Theorem 22.1.

(b)

Show that $\mathbb {P}\left(n^{-1} \max_{k \leq n} X_{k} \leq x\right) \rightarrow e^{-u / \pi x}$ for $x>0$ . Relate to Theorem 14.3.

Problem 22.6

If $X_{1}, X_{2}, \ldots$ are independent and identically distributed, and if $\mathbb {P}\left(X_{1} \geq 0\right)=1$ and $\mathbb {P}\left(X_{1}>0\right)>0$ , then $\sum_{n} X_{n}=\infty$ with probability 1. Deduce this from Theorem 22.1 and its corollary and also directly: find a positive $\epsilon$ such that $X_{r}>c$ infinitely often with probability 1.

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Problem 22.7

Suppose that $X_{1}, X_{2}, \ldots$ are independent and identically distributed and $\mathbb {E}\left[\left|X_{1}\right|\right]=\infty$ . Use (21.9) to show that $\sum_{n} \mathbb {P}\left(\left|X_{n}\right| \geq a n\right)=\infty$ for each $a$ , and conclude that $\sup_{n} n^{-1}\left|X_{n}\right|=\infty$ with probability 1. Now show that $\sup_{n} n^{-1}\left|S_{n}\right| =\infty$ with probability 1. Compare with the corollary to Theorem 22.1.

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Problem 22.8

(a)(b)

Wald's equation. Let $X_{1}, X_{2}, \ldots$ be independent and identically distributed with finite mean, and put $S_{n}=X_{1}+\cdots +X_{n}$ . Suppose that $\tau$ is a stopping time: $\tau$ has positive integers as values and $[\tau =n] \in \sigma \left(X_{1}, \ldots , X_{n}\right)$ ; see Section 7 for examples. Suppose also that $\mathbb {E}\left[\tau \right]<\infty$ .

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(a)

Prove that

\mathbb {E}\left[S_{\tau }\right]=\mathbb {E}\left[X_{1}\right] \mathbb {E}\left[\tau \right]. \tag {22.21}

(b)

Suppose that $X_{n}$ is $\pm 1$ with probabilities $p$ and $q, p \neq q$ , let $\tau$ be the first $n$ for which $S_{n}$ is $-a$ or $b$ ( $a$ and $b$ positive integers), and calculate $\mathbb {E}\left[\tau \right]$ . This gives the expected duration of the game in the gambler's ruin problem for unequal $p$ and $q$ .

Problem 22.9

$20.9 \uparrow$ Let $Z_{n}$ be 1 or 0 according as at time $n$ there is or is not a record in the sense of Problem 20.9. Let $R_{n}=Z_{1}+\cdots +Z_{n}$ be the number of records up to time $n$ . Show that $R_{n} / \log n \rightarrow_{P} 1$ .

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Problem 22.10

(a)(b)

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(a)

$22.1 \uparrow$ Show that for an independent sequence $\left\{ X_{n}\right\}$ the radius of convergence of the random Taylor series $\sum_{n} X_{n} z^{n}$ is $r$ with probability 1 for some nonrandom $r$ .

(b)

Suppose that the $X_{n}$ have the same distribution and $\mathbb {P}\left(X_{1} \neq 0\right)>0$ . Show that $r$ is 1 or 0 according as $\log^{+}\left|X_{1}\right|$ has finite mean or not.

Problem 22.11

Suppose that $X_{0}, X_{1}, \ldots$ are independent and each is uniformly distributed over $[0,2 \pi ]$ . Show that with probability 1 the series $\sum_{n} e^{i X_{n}} z^{n}$ has the unit circle as its natural boundary.

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Problem 22.12

Prove (what is essentially Kolmogorov's zero-one law) that if $A$ is independent of a $\pi$ -system $\mathscr {P}$ and $A \in \sigma (\mathscr {P})$ , then $\mathbb {P}\left(A\right)$ is either 0 or 1.

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Problem 22.13

Suppose that $\mathscr {A}$ is a semiring containing $\Omega$ .

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(a)

Show that if $\mathbb {P}\left(A \cap B\right) \leq b \mathbb {P}\left(B\right)$ for all $B \in \mathscr {A}$ , and if $b<1$ and $A \in \sigma (\mathscr {A})$ , then $\mathbb {P}\left(A\right)=0$ .

(b)

Show that if $\mathbb {P}\left(A \cap B\right) \leq \mathbb {P}\left(A\right) \mathbb {P}\left(B\right)$ for all $B \in \mathscr {A}$ , and if $A \in \sigma (\mathscr {A})$ , then $\mathbb {P}\left(A\right)$ is 0 or 1.

(c)

Show that if $a \mathbb {P}\left(B\right) \leq \mathbb {P}\left(A \cap B\right)$ for all $B \in \mathscr {A}$ , and if $a>0$ and $A \in \sigma (\mathscr {A})$ , then $\mathbb {P}\left(A\right)=1$ .

(d)

Show that if $\mathbb {P}\left(A\right) \mathbb {P}\left(B\right) \leq \mathbb {P}\left(A \cap B\right)$ for all $B \in \mathscr {A}$ , and if $A \in \sigma (\mathscr {A})$ , then $\mathbb {P}\left(A\right)$ is 0 or 1.

(e)

Reconsider Problem 3.20

Problem 22.14

(a)(b)(c)

$22.12 \uparrow$ Burstin's theorem. Let $f$ be a Borel function on $[0,1]$ with arbitrarily small periods: For each $\epsilon$ there is a $p$ such that $0<p<\epsilon$ and $f(x)=f(x+p)$ for $0 \leq x \leq 1-p$ . Show that such an $f$ is constant almost everywhere:

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(a)

Show that it is enough to prove that $\mathbb {P}\left(f^{-1} B\right)$ is 0 or 1 for every Borel set $B$ , where $P$ is Lebesgue measure on the unit interval.

(b)

Show that $f^{-1} B$ is independent of each interval $[0, x]$ , and conclude that $\mathbb {P}\left(f^{-1} B\right)$ is 0 or 1.

(c)

Show by example that $f$ need not be constant.

Problem 22.15

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

Assume that $X_{1}, \ldots , X_{n}$ are independent and $s, t, \alpha$ are nonnegative Let

\begin{aligned} L(s) & =\max _{k \leq n} \mathbb {P}\left(\left|S_{k}\right| \geq s\right), R(s)=\max _{k \leq n} \mathbb {P}\left(\left|S_{n}-S_{k}\right|>s\right), \\ M(s) & =\mathbb {P}\left(\max _{k \leq n}\left|S_{k}\right| \geq s\right), T(s)=\mathbb {P}\left(\left|S_{n}\right| \geq s\right). \end{aligned}

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(a)

Following the first part of the proof of (22.10), show that

M(s+t) \leq T(t)+M(s+t) R(s) \tag {22.22}

(b)

Take $s=2 \alpha$ and $t=\alpha$ ; use (22.22), together with the inequalities $T(s) \leq L(s)$ and $R(2 s) \leq 2 L(s)$ , to prove Etemadi's inequality (22.10) in the form

M(3 \alpha ) \leq B_{E}(\alpha )=1 \wedge 3 L(\alpha ). \tag {22.23}

(c)

Carry the rightmost term in (22.22) to the left side, take $s=t=\alpha$ , and prove Ottaviani's inequality:

M(2 \alpha ) \leq B_{O}(\alpha )=1 \wedge \frac{T(\alpha )}{1-R(\alpha )} \tag {22.24}

(d)

Prove

B_{E}(\alpha ) \leq 3 B_{O}(\alpha / 2), \quad B_{O}(\alpha ) \leq 3 B_{E}(\alpha / 6).

This shows that the Etemadi and Ottaviani inequalities are of the same power for most purposes (as, for example, for the proofs of Theorem 22.7 and (37.9)). Etemadi's inequality seems the more natural of the two. Neither inequality can replace (9.39) in the proof of the law of the iterated logarithm.