Math Problems Vault

Problem 6.1

Establish the following generalized version of Theorem 2 (the Hölder inequality). Namely, show that for any $n( \geq 2)$ r.v.s $X_{1}, \ldots , X_{n}$ and any positive numbers $r_{1}, \ldots , r_{n}$ with $\frac{1}{r_{1}}+\cdots +\frac{1}{r_{n}} = 1$ , it holds that

\mathbb {E}\left[\left|X_{1} \ldots X_{n}\right|\right] \leq \mathbb {E}\left[\left|X_{1}\right|^{r_{1}}\right]^{\frac{1}{r_{1}}} \ldots \mathbb {E}\left[\left|X_{n}\right|^{r_{n}}\right]^{\frac{1}{r_{n}}} .

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Problem 6.2

(i)(ii)

Let $g: I$ open interval in $\mathbb {R} \rightarrow \mathbb {R}$ be convex; i.e., for every $x, x^{\prime } \in I$ and every $\alpha \in [0,1]$ , it holds $g\left(\alpha x+(1-\alpha ) x^{\prime }\right) \leq \alpha g(x)+(1-\alpha ) g\left(x^{\prime }\right)$ . Then show that

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(i)

$g$ is continuous.

(ii)

For every $x_{0} \in I$ , there exists $\lambda \left(x_{0}\right) \in \mathbb {R}$ such that $g(x)-g\left(x_{0}\right) \geq \lambda \left(x_{0}\right) \times \left(x-x_{0}\right), x \in I$ .

Problem 6.3

Establish the following generalized version of Theorem 3 (the Minkowski inequality). Namely, show that for any $n( \geq 2)$ r.v.s $X_{1}, \ldots , X_{n}$ and any $r \geq 1$ , it holds that

\mathbb {E}\left[\left|X_{1}+\cdots +X_{n}\right|^{r}\right]^{\frac{1}{r}} \leq \mathbb {E}\left[\left|X_{1}\right|^{r}\right]^{\frac{1}{r}}+\cdots +\mathbb {E}\left[\left|X_{n}\right|^{r}\right]^{\frac{1}{r}} .

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Problem 6.4

If $\sum_{n = 1}^{\infty } \mathbb {E}\left[\left|X_{n}-X\right|^{r}\right] < \infty$ for some $r > 0$ , then $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} X$ .

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Problem 6.5

If $X_{n} \xrightarrow [ \rightarrow \infty ]{\text{ a.s. }} X$ , and $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} Y$ for some $r > 0$ , it follows that $\mathbb {P}\left(X \neq Y\right) = 0$ .

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Problem 6.6

Construct r.v.s $X_{n}, n \geq 1$ , and $X$ on some probability space ( $\Omega , \mathcal{A}, P$ ) such that $X_{n} \xrightarrow [n \rightarrow \infty ]{P} 0$ , but $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} \nrightarrow 0$ for any $r \geq 1$ .

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Problem 6.7

Construct r.v.s $X_{n}, n \geq 1$ , and $X$ on some probability space ( $\Omega , \mathcal{A}, P$ ) such that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} \nrightarrow X$ , but $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} X$ for any $r > 0$ .

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Problem 6.8

Let $X_{n}, n = 1,2, \ldots$ , be r.v.s such that $\mathbb {E}\left[\left|X_{n}\right|^{\beta }\right] \leq M( < \infty ), n \geq 1$ , for some $\beta > 0$ . Then show that $\left|X_{n}\right|^{\alpha }, n \geq 1$ , are uniformly integrable for all $\alpha$ with $0 < \alpha < \beta$ .

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Problem 6.9

For $n = 1,2, \ldots$ , let $X_{n}$ be r.v.s such that $\mathbb {P}\left(X_{n} = c n\right) = \frac{1}{n}$ and $\mathbb {P}\left(X_{n} = 0\right) = 1-\frac{1}{n}$ , for some $c > 0$ . Investigate whether or not these r.v.s are uniformly integrable.

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Problem 6.10

If $\mathbb {E}\left[\left|X\right|\right] < \infty$ , show that $\int \left|X\right|$ is absolutely continuous; that is, $\int_{A}\left|X\right| \rightarrow 0$ when $\mathbb {P}\left(A\right) \rightarrow 0$ .

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Problem 6.11

For $n = 1,2, \ldots$ , let $X_{n}, Y_{n}$ and $X, Y$ be r.v.s such that $\mathbb {P}\left(X_{n} \geq Y_{n} \geq 0\right) = 1$ , $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X, Y_{n} \xrightarrow [n \rightarrow \infty ]{P} Y$ , and $\mathbb {E}\left[X_{n}\right] \underset {n \rightarrow \infty }{\rightarrow } \mathbb {E}\left[X\right]$ finite. Then show that $\mathbb {E}\left[\left|Y_{n}-Y\right|\right] \underset {n \rightarrow \infty }{\rightarrow } 0$ .

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Problem 6.12

For any r.v. $X$ defined on the measure space $(\Omega , \mathcal{A}, \mu )$ , show directly that the Markov inequality holds; namely, $\mu (\left|X\right| \geq c) \leq c^{-r} \int \left|X\right|^{r} d \mu$ for any $r > 0$ and any $c > 0$ .

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Problem 6.13

For a r.v. $X$ and some $r > 0$ , show that $\mathbb {E}\left[\left|X\right|^{r}\right] < \infty$ if and only if $\mathbb {E}\left[\left|X-c\right|^{r}\right] < \infty$ for every (finite) constant $c$ .

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Problem 6.14

For a r.v. $X$ with $\mathbb {E}\left[X\right] = \mu \in \mathbb {R}$ and $\operatorname {Var}\left[X\right] = \sigma^{2} = 0$ , show that $\mathbb {P}\left(X = \mu \right) = 1$ .

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Problem 6.15

(i)(ii)

For $n = 1,2, \ldots$ , consider the r.v.s $X_{n}$ and $X$ and show that

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(i)

$\left|X_{n}\right| \underset {n \rightarrow \infty }{P} 0$ if and only if $\frac{\left|X_{n}\right|}{1+\left|X_{n}\right|} \xrightarrow [n \rightarrow \infty ]{P} 0$ .

(ii)

$\left|X_{n}\right| \xrightarrow [n \rightarrow \infty ]{P} 0$ if and only if $\mathbb {E}\left[\frac{\left|X_{n}\right|}{1+\left|X_{n}\right|}\right] \underset {n \rightarrow \infty }{\rightarrow } 0$ .

Problem 6.16

(i)(ii)

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(i)

For $r > 1$ and $x_{j} \in \mathbb {R}, j = 1, \ldots , n$ , show that

\left|n^{-1} \sum _{j = 1}^{n} x_{j}\right|^{r} \leq n^{-1} \sum _{j = 1}^{n}\left|x_{j}\right|^{r} .

(ii)

From part (i), deduce that

\mathbb {E}\left[\left|\frac{1}{n} \sum _{j = 1}^{n} X_{j}\right|^{r}\right] \leq \frac{1}{n} \sum _{j = 1}^{n} \mathbb {E}\left[\left|X_{j}\right|^{r}\right] .

(On the other hand, by the Minkowski inequality,

\mathbb {E}\left[\left|\frac{1}{n} \sum _{j = 1}^{n} X_{j}\right|^{r}\right]^{1 / r} \leq \frac{1}{n} \sum _{j = 1}^{n} \mathbb {E}\left[\left|X_{j}\right|^{r}\right]^{1 / r} .

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Problem 6.17

For $n = 1,2, \ldots$ , suppose that $X_{n}$ and $X$ are r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$ , and suppose that $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$ and $\left\{ X_{n}\right\}$ are uniformly integrable. Then show that $\int_{A} X_{n} d P \underset {n \rightarrow \infty }{\rightarrow } \int_{A} X d P$ uniformly in $A \in \mathcal{A}$ .

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Problem 6.18

If for $n = 1,2, \ldots$ , the r.v.s $\left\{ X_{n}\right\}$ and $\left\{ Y_{n}\right\}$ are uniformly integrable, show that $\left\{ X_{n}+Y_{n}\right\}$ are also uniformly integrable.

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Problem 6.19

For $n = 1,2, \ldots$ , let $X_{n}$ be identically distributed with $\mathbb {E}\left[X_{n}\right] \in \mathbb {R}$ . Then show that $\left\{ \bar{X}_{n}\right\}$ are uniformly integrable, where $\bar{X}_{n} = \frac{1}{n} \sum_{j = 1}^{n} X_{j}$ is the sample mean of $X_{1}, \ldots , X_{n}$ .

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Problem 6.20

Let $Y = \sup_{n \geq 1}\left|X_{n}\right|$ with $\mathbb {E}\left[Y^{r}\right] < \infty$ (for some $r > 0$ ) and $X_{n} \xrightarrow [n \rightarrow \infty ]{\stackrel{\text{ a.s. }}{\rightarrow }} X$ . Then show that $\mathbb {E}\left[\left|X\right|^{r}\right] < \infty$ and $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} X$ .

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Problem 6.21

For $n = 1,2, \ldots$ , let $X_{n}$ be r.v.s such that $\mathbb {E}\left[X_{n}\right] = \mu_{n} \in \mathbb {R}$ and $\operatorname {Var}\left[X_{n}\right] = \sigma_{n}^{2} \underset {n \rightarrow \infty }{\rightarrow } 0$ . Then show that $X_{n}-\mu_{n} \xrightarrow [n \rightarrow \infty ]{P} 0$ .

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Problem 6.22

Let the r.v.s $X_{n}, n \geq 1$ , be defined as follows: $X_{n}(\omega ) = 2^{n}$ if $\omega \in \left(0, \frac{1}{n}\right)$ , and $X_{n}(\omega ) = 0$ otherwise, where $(\Omega , \mathcal{A}, P) = \left((0,1], \mathcal{B}_{(0,1]}, \lambda \right)$ and $\lambda$ is the Lebesgue measure. Then show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\rightarrow } 0$ pointwise, but $\mathbb {E}\left[\left|X_{n}\right|^{r}\right] \underset {n \rightarrow \infty }{\rightarrow }$ to any finite number for any $r > 0$ ; in fact, $\mathbb {E}\left[\left|X_{n}\right|^{r}\right] \underset {n \rightarrow \infty }{\rightarrow } \infty$ for any $r > 0$ .

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Problem 6.23

(i)(ii)(iii)

For $n = 1,2, \ldots$ , let the r.v. $X_{n}$ be defined by

X_{n} = \left\{ \begin{array}{l} n^{c} \text{ with probability } \frac{1}{n} \\ 0 \text{ with probability } 1-\frac{2}{n} \\ n^{-c} \text{ with probability } \frac{1}{n}, \quad \text{ where } c \text{ is a positive constant. } \end{array}\right.

Then show that

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(i)

$X_{n} \xrightarrow [n \rightarrow \infty ]{P} 0$ .

(ii)

$\mathbb {E}\left[\left|X_{n}\right|^{r}\right] \underset {n \rightarrow \infty }{\rightarrow } 0,0 < c r < 1 ; \mathbb {E}\left[\left|X_{n}\right|^{r}\right] \underset {n \rightarrow \infty }{\rightarrow } \infty , c r > 1$ .

(iii)

$X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} 0, c r < 1$ .

Problem 6.24

Let $\mathbb {E}\left[X_{n}^{2}\right] < \infty$ for all $n$ . Then, if $\mathbb {E}\left[\left|X_{n}-X\right|^{2}\right] \underset {n \rightarrow \infty }{\rightarrow } 0$ , show that $\mathbb {E}\left[\left|X_{n}^{2}-X^{2}\right|\right] \underset {n \rightarrow \infty }{\rightarrow } 0$ .

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Problem 6.25

For $n = 1,2, \ldots$ , let $X_{n}, Y_{n}$ and $X, Y$ be r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$ , and suppose that $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} X, Y_{n} \xrightarrow [n \rightarrow \infty ]{(s)} Y$ where $r, s > 1$ with $\frac{1}{r}+ \frac{1}{s} = 1$ . Then show that $X_{n} Y_{n} \xrightarrow [n \rightarrow \infty ]{(1)} X Y$ .

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Problem 6.26

Let $(\Omega , \mathcal{A}, P) = \left((0,1), \mathcal{B}_{(0,1)}, \lambda \right)$ , where $\lambda$ is the Lebesgue measure, and let $X_{n}$ be a r.v. defined by

X_{n} = \left\{ \begin{array}{l} 1, \omega \in \left(0, \frac{1}{n}\right) \\ 0, \omega \in \left[\frac{1}{n}, 1\right) \end{array}, \quad n = 1,2, \ldots \right.

Then show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\stackrel{(r)}{\rightarrow }} 0$ for any $r > 0$ , and $X_{n} \xrightarrow [n \rightarrow \infty ]{\stackrel{\text{ a.s. }}{\rightarrow }} 0$ ; indeed, $X_{n} \xrightarrow [n \rightarrow \infty ]{\longrightarrow } 0$ pointwise. (Compare it with Exercise 7.)

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Problem 6.27

If $X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} X$ , show that there is a subsequence $\left\{ X_{n_{k}}\right\} \subseteq \left\{ X_{n}\right\}$ such that $X_{n_{k}} \xrightarrow {\xrightarrow {\text{ a.s. }}} X$ .

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Problem 6.28

(i)(ii)

For $n = 1,2, \ldots$ , let the r.v. $X_{n}$ be defined by

X_{n} = \left\{ \begin{array}{l} e^{n} \text{ with probability } n^{-2} \\ 0 \text{ with probability } 1-n^{-2} \end{array}\right.

Then show that

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(i)

$X_{n} \xrightarrow [n \rightarrow \infty ]{P} 0$ (indeed, $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} 0$ ).

(ii)

$X_{n} \xrightarrow [n \rightarrow \infty ]{(r)} \nrightarrow 0$ for any $r > 0$ (in fact, $\mathbb {E}\left[\left|X_{n}\right|^{r}\right] \xrightarrow [n \rightarrow \infty ]{\longrightarrow } \infty$ for all $r > 0$ ). (Compare it with Exercises 6 and Theorem 8.)

Problem 6.29

(i)(ii)

For $n = 1,2, \ldots$ , let the r.v. $X_{n}$ be defined by

X_{n} = \begin{cases} 2^{c n} & \text{ with probability } 2^{-n} \\ 0 & \text{ with probability } 1-2^{-n+1} \\ -2^{c n} & \text{ with probability } 2^{-n}, \text{ for some positive constant } c . \end{cases}

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(i)

Show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} 0$ .

(ii)

Determine the condition $c$ and $r( > 0)$ must satisfy, so that $X \underset {n \rightarrow \infty }{\stackrel{(r)}{\rightarrow }} 0$ . (Compare it with Exercises 6 and 8.)

Problem 6.30

(i)(ii)(iii)

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(i)

For any r.v. $X$ , show that $\mathbb {P}\left(\left|X\right| \geq c\right) \leq \mathbb {E}\left[\left|X\right|\right] / c(c > 0)$ , by using the obvious inequality $\left|X\right| \geq c I(\left|X\right| \geq c)$ . In particular, if $X \geq 0$ , then $\mathbb {P}\left(X \geq c\right) \leq \mathbb {E}\left[X\right] / c$ .

(ii)

For any r.v. $X$ and any $c > 0$ , show that

\mathbb {P}\left(X \geq c\right) \leq e^{-t c} \mathbb {E}\left[e^{t X}\right](t > 0), \mathbb {P}\left(X \leq c\right) \leq e^{-t c} \mathbb {E}\left[e^{t X}\right](t < 0) .

(iii)

If $\mathbb {E}\left[X^{2}\right] < \infty$ , then show that $\mathbb {P}\left(\left|X\right| > 0\right) \geq (\mathbb {E}\left[\left|X\right|\right])^{2} / \mathbb {E}\left[X^{2}\right]$ . In particular, if $X \geq 0$ , then $\mathbb {P}\left(X > 0\right) \geq (\mathbb {E}\left[X\right])^{2} / \mathbb {E}\left[X^{2}\right]$ .

Problem 6.31

For $n \geq 1$ , let $X_{n}$ be r.v.s with $\mathbb {E}\left[X_{n}\right] = \mu_{n}, \sigma^{2}\left(X_{n}\right) = \sigma_{n}^{2}$ , and suppose that, as $n \rightarrow \infty , \mu_{n} \rightarrow \mu \in \mathbb {R}$ , and $\sigma_{n}^{2} \rightarrow 0$ . Then show that $\mathbb {E}\left[X_{n}-\mu^{2}\right] \rightarrow 0$ and $X_{n} \xrightarrow {P} \mu$ .

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Problem 6.32

(i)(ii)

For $n \geq 1$ , let $X_{n}$ and $Y_{n}$ be r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$ , and suppose that $\left|X_{n}\right|$ and $\left|Y_{n}\right|, n \geq 1$ , are uniformly integrable and $X_{n}-Y_{n} \xrightarrow {P} 0$ as $n \rightarrow \infty$ . Then show that:

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(i)

$\left|X_{n}-Y_{n}\right|, n \geq 1$ , are uniformly integrable.

(ii)

$\mathbb {E}\left[\left|X_{n}-Y_{n}\right|\right] \underset {n \rightarrow \infty }{\longrightarrow } 0$ .

Problem 6.33

(i)(ii)

Consider the probability space $(\Omega , \mathcal{A}, P)$ , and let $\Theta$ be an open subset of $\mathbb {R}$ . Let $g(\cdot ; \cdot ): \Omega \times \Theta \rightarrow \mathbb {R}$ be $\left(\mathcal{A} \times \mathcal{B}_{\Theta }\right)$ -measurable, where $\mathcal{B}_{\Theta }$ is the $\sigma$ -field of Borel subsets of $\Theta$ . We say that $g(\cdot ; \theta )$ is differentiable in q.m. at $\theta$ , if there exists a $\left(\mathcal{A} \times \mathcal{B}_{\Theta }\right)$ -measurable function $\dot{g}(\cdot ; \theta )$ (the quadratic mean derivative of $g(\cdot ; \theta )$ at $\theta$ ) such that

h^{-1}[g(\cdot ; \theta +h)-g(\cdot ; \theta )] \underset {n \rightarrow \infty }{\text{ q.m. }} \dot{g}(\cdot ; \theta ) .

Now let $p(x ; \theta ) = \frac{1}{2} e^{-\left|x-\theta \right|}, x \in \mathbb {R}(\theta \in \mathbb {R})$ be the double exponential probability density function, and for $\theta , \theta^{*} \in \mathbb {R}$ , set

g\left(x ; \theta , \theta ^{*}\right) = \left[p\left(x ; \theta ^{*}\right) / p(x ; \theta )\right]^{1 / 2} = \exp \left(\frac{1}{2}\left|x-\theta \right|-\frac{1}{2}\left|x-\theta ^{*}\right|\right) .

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(i)

Show that $g\left(x ; \theta , \theta^{*}\right)$ is not pointwise differentiable with respect to $\theta^{*}$ at $(\theta , \theta )$ for any $\theta = x \in \mathbb {R}$ .

(ii)

If the r.v. $X$ is distributed according to $p(\cdot ; \theta )$ , show that $g\left(X ; \theta , \theta^{*}\right)$ is differentiable in q.m. with respect to $\theta^{*}$ at $(\theta , \theta ), \theta \in \mathbb {R}$ , with q.m. derivative $\dot{g}(X ; \theta )$ given by:

\dot{g}(X ; \theta ) = \left\{ \begin{array}{cl} -1 / 2 & \text{ if } X < \theta \\ 0 \text{ (for example) } & \text{ if } X = \theta \\ 1 / 2 & \text{ if } X > \theta \end{array}\right.

Problem 6.34

(i)(ii)(iii)(iv)

All r.v.s appearing below are defined on the probability space $(\Omega , \mathcal{A}, P)$ .

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(i)

If $X \geq 0$ , then $\mathbb {E}\left[X\right] \geq 0$ and $\mathbb {E}\left[X\right] = 0$ only if $\mathbb {P}\left(X = 0\right) = 1$ .

(ii)

Let $X \geq Y$ with finite expectations. Then $\mathbb {E}\left[X\right] \geq \mathbb {E}\left[Y\right]$ and $\mathbb {E}\left[X\right] = \mathbb {E}\left[Y\right]$ only if $\mathbb {P}\left(X = Y\right) = 1$ .

(iii)

Let $X > Y$ with finite expectations. Then $\mathbb {E}\left[X\right] > \mathbb {E}\left[Y\right]$ .

(iv)

Let $g: I$ open subset of $\mathbb {R}$ be strictly convex (i.e.,

g\left[\alpha x+(1-\alpha ) x^{\prime }\right] < \alpha g(x)+(1-\alpha ) g\left(x^{\prime }\right),

$x, x^{\prime } \in I, 0 < \alpha < 1$ ), let $Z$ be a r.v. taking values in $I$ , let $\mathbb {E}\left[Z\right] \in I$ , and let $\mathbb {E}\left[g(Z)\right]$ exist. Then $\mathbb {E}\left[g(Z)\right] > g(\mathbb {E}\left[Z\right])$ unless $\mathbb {P}\left(Z =$ constant $\right) = 1$ .