Math Problems Vault

Problem 7.1

Let $B$ be a subset of a metric space. For each pair of points $x_{1}$ and $x_{2}$ in $X$ , show that $\inf_{y \in B} d\left(x_{1}, y\right) \leq d\left(x_{1}, x_{2}\right)+\inf_{y \in B} d\left(x_{2}, y\right)$ . Deduce that the function $f_{B}(x): = d(x, B)$ satisfies the Lipschitz condition $\left|f_{B}(x)-f_{B}(y)\right| \leq d(x, y)$ .

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Problem 7.2

For real-valued functions $f, g$ on $X$ , prove that $\left\| f g\right\|_{B L} \leq 2\left\| f\right\|_{B L}\left\| g\right\|_{B L}$ .

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Problem 7.3

Let $B$ be a subset of a metric space $X$ . Show that

\begin{aligned} \left\{ x: d(x, B) = 0\right\} & = \bar{B}: = \text{ closure of } B, \\ \left\{ x: d\left(x, B^{c}\right) > 0\right\} & = \stackrel{\circ }{B}: = \text{ interior of } B, \\ \left\{ x: d(x, B) = 0 = d\left(x, B^{c}\right)\right\} & = \bar{B} \backslash B = \partial B = \text{ boundary of } B . \end{aligned}

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Problem 7.4

(i)(ii)(iii)

Let $P$ be a probability measure on a separable metric space, and $B$ be a Borel set for which $P(\partial B) > 0$ .

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(i)

For each $\epsilon > 0$ , show that there is a partition of $\partial B$ into disjoint Borel sets $\left\{ D_{i}: i \in \mathbb {N}\right\}$ each with diameter less than $\epsilon$ . Hint: Consider the union of balls of radius $\epsilon / 3$ centered at the points of a countable dense subset of $\partial B$ .

(ii)

For each $i$ , find points $x_{i} \in B$ and $y_{i} \in B^{c}$ such that $d\left(x_{i}, D_{i}\right) < \epsilon$ and $d\left(y_{i}, D_{i}\right) < \epsilon$ . Define a probability measure $P_{\epsilon }^{\prime }$ by replacing all the $P$ mass in $D_{i}$ by a point mass (of size $P D_{i}$ ) at $x_{i}$ , for each $i$ . Define $P_{\epsilon }^{\prime \prime }$ similarly, by concentrating the mass in each $D_{i}$ at $y_{i}$ . Show that $P_{\epsilon }^{\prime } B = P \bar{B}$ and $P_{\epsilon }^{\prime \prime } B = P \stackrel{\circ }{B}$ for each $\epsilon > 0$ .

(iii)

Show that, for each $\ell$ with $\left\| \ell \right\|_{B L}: = K < \infty$ ,

\left|P \ell -P_{\epsilon }^{\prime } \ell \right| \leq K \epsilon P(\partial B) \quad \text{ and } \quad \left|P \ell -P_{\epsilon }^{\prime \prime } \ell \right| \leq K \epsilon P(\partial B) .

Deduce that $P_{\epsilon }^{\prime } \leadsto P$ and $P_{\epsilon }^{\prime \prime } \leadsto P$ as $\epsilon \rightarrow 0$ , even though we have $P_{\epsilon }^{\prime } B \equiv P \bar{B} > P B$ and $P_{\epsilon }^{\prime \prime } B \equiv P \stackrel{\circ }{B} < P B$ .

Problem 7.5

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

For $\mathbf{x} = \left(x_{1}, x_{2}\right) \in \mathbb {R}^{2}$ , define $Q_{\mathrm{x}} = \left\{ \left(y_{1}, y_{2}\right) \in \mathbb {R}^{2}: y_{1} \leq x_{1}, y_{2} \leq x_{2}\right\}$ , the quadrant with vertex x . Let $\left\{ X_{n}\right\}$ be a sequence of random vectors and $P$ be a probability measure such that $\mathbb {P}\left(X_{n} \in Q_{\mathrm{x}}\right) \rightarrow P Q_{\mathrm{x}}$ for all x such that $P\left(\partial Q_{\mathrm{x}}\right) = 0$ . Show that $X_{n} \rightsquigarrow P$ by following these steps.

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(i)

Write $\mathcal{L}$ for the class of all lines parallel to a coordinate axis. Show that all except countably many lines in $\mathcal{L}$ have zero $P$ measure. (Hint: How many horizontal lines can have $P$ measure greater than $1 / n$ ?)

(ii)

Given $\ell \in B L\left(\mathbb {R}^{2}\right)$ , with $0 \leq \ell \leq 1$ , show that there exist disjoint rectangles $S_{1}, \ldots , S_{m}$ with sides parallel to the coordinate axes, such that (a) $P\left(\partial S_{i}\right) = 0$ for each each $i$ ; (b) the set $B = \bigcup_{i} S_{i}$ has $P$ -measure greater than $1-\epsilon$ ; (c) the function $\ell$ oscillates by less than $\epsilon$ on each $S_{i}$ .

(iii)

Choose arbitrarily points $\mathbf{x}_{i}$ in $S_{i}$ , for $i = 1, \ldots , m$ . Define functions $g_{\epsilon }(\mathbf{x}) = \sum_{i}\left\{ \mathbf{x} \in S_{i}\right\} \ell \left(\mathbf{x}_{i}\right)$ . Show that $\left|g_{\epsilon }(\mathbf{x})-\ell (\mathbf{x})\right| \leq \epsilon +\left\{ \mathbf{x} \notin B\right\}$ for all $\mathbf{x}$ .

(iv)

Use a sandwiching argument to show that $\mathbb {E}\left[\ell \left(X_{n}\right)\right] \rightarrow P \ell$ , then deduce that $X_{n} \leadsto P$ .

Problem 7.6

Let $X$ be a metric space. Show that the map $\psi :(x, y) \rightarrow d(x, y)$ is continuous. Deduce that $\psi$ is $\mathcal{B}\left(X^{2}\right) \backslash \mathcal{B}(\mathbb {R})$ -measurable. If $X$ is separable, deduce that $\psi$ is $\mathcal{B}(X) \otimes \mathcal{B}(X) \backslash \mathcal{B}(\mathbb {R})$ -measurable, and hence $\omega \mapsto d(X(\omega ), Y(\omega ))$ is measurable if $X$ and $Y$ are random elements of $X$ .

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Problem 7.7

If $P \ell = Q \ell$ for each $\ell$ in $B L(X)$ , show that $P = Q$ as measures on $\mathcal{B}(X)$ .

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Problem 7.8

Let $X$ be a separable metric space. Show that $\Delta (P, Q): = \sup \left\{ \left|P \ell -Q \ell \right|:\left\| \ell \right\|_{B L} \leq 1\right\}$ is a metric for weak convergence. That is, show that $P_{n} \leadsto P$ if and only if $\Delta \left(P_{n}, P\right) \rightarrow 0$ , by following these steps. (Read the proof of Lemma $\langle 37 \rangle$ for hints.)

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(i)

Show that $\Delta$ is a metric, and $\left|P \ell -Q \ell \right| \leq \left\| \ell \right\|_{B L} \Delta (Q, P)$ , for each $\ell$ . Deduce that $P_{n} \ell \rightarrow P \ell$ , for each $\ell \in B L(X)$ , if $D\left(P_{n}, P\right) \rightarrow 0$ .

(ii)

Let $X_{0}: = \left\{ x_{i}: i \in \mathbb {N}\right\}$ be dense in $X$ . For a fixed $\epsilon > 0$ , define $f_{0}(x) \equiv \epsilon$ and $f_{i}(x): = \left(1-d\left(x, x_{i}\right) / \epsilon \right)^{+}$ . Define $G_{k}: = \bigcup_{i = 1}^{k}\left\{ f_{i} > 1 / 2\right\}$ . Choose $k$ so that $P G_{k}^{k} < \epsilon$ . Show that each function $\ell_{i}: = f_{i} / \sum_{0 \leq i \leq k} f_{i}$ belongs to $B L(X)$ . Show that $\ell_{0}(x) \leq 2 \epsilon$ for $x \in G_{k}$ . Show that $\sum_{i = 0}^{k} \ell_{i} \equiv 1$ . Show that $\operatorname {diam}\left\{ \ell_{i} > 0\right\} \leq 2 \epsilon$ for $i \geq 1$ .

(iii)

For each $i \geq 1$ , choose an $x_{i}$ from $\left\{ \ell_{i} > 0\right\}$ . Show that

\left|\ell (x)-\sum _{i = 1}^{k} \ell \left(x_{i}\right) \ell _{i}(x)\right| \leq 4 \epsilon +G_{k}^{c} \quad \text{ if }\left\| \ell \right\| _{B L} \leq 1.

(iv)

For each probability measure $Q$ , and each $\ell$ with $\left\| \ell \right\|_{B L} \leq 1$ , show that

\left|Q \ell -P \ell \right| \leq 8 \epsilon +Q G_{k}^{c}+P G_{k}^{c}+\sum _{i = 1}^{k}\left|Q \ell _{i}-P \ell _{i}\right|

(v)

If $P_{n} \rightsquigarrow P$ , deduce that $\Delta \left(P_{n}, P\right) < 10 \epsilon$ eventually.

Problem 7.9

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

Let $y$ and $z$ be metric spaces, such that $y$ is separable. Let $d$ be the metric on $y \otimes z$ defined in Example $\langle 14 \rangle$ .

Example 14: For metric spaces $\left(y, d_{y}\right)$ and $\left(\mathcal{Z}, d_{\mathcal{Z}}\right)$ , equip $X: = y \times \mathcal{Z}$ with the metric $d\left(x_{1}, x_{2}\right): = \max \left(d_{y}\left(y_{1}, y_{2}\right), d_{\mathcal{Z}}\left(z_{1}, z_{2}\right)\right)$ , where $x_{i}: = \left(y_{i}, z_{i}\right)$ .

Let $P_{0}, P$ be probability measures on $\mathcal{B}(y)$ , and $Q_{0}, Q$ be probability measure on $\mathcal{B}(\mathcal{Z})$ . For a fixed $\ell$ in $B L(y \times \mathcal{Z})$ , define $h_{0}(z): = P_{0}^{y} \ell (y, z)$ and $g_{Q}(y): = Q^{z} \ell (y, z)$ .

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(i)

Show that $\left\| h_{0}\right\|_{B L} \leq \left\| \ell \right\|_{B L}$ and $\left\| g_{Q}\right\|_{B L} \leq \left\| \ell \right\|_{B L}$ .

(ii)

Let $\Delta_{y}$ and $\Delta_{z}$ denote the analogs of the metric from Problem [8]. Show that

\begin{aligned} \left|(P \otimes Q) \ell -\left(P_{0} \otimes Q_{0}\right) \ell \right| & \leq \left|P g_{Q}-P_{0} g_{Q}\right|+\left|Q h_{0}-Q_{0} h_{0}\right| \\ & \leq \left\| \ell \right\| _{B L} \Delta y\left(P, P_{0}\right)+\left|Q h_{0}-Q_{0} h_{0}\right| . \end{aligned}

(iii)

Show that $\Delta_{y_{x} z}\left(P \otimes Q, P_{0} \otimes Q_{0}\right) \leq \Delta_{y}\left(P, P_{0}\right)+\Delta_{z}\left(Q, Q_{0}\right)$ .

(iv)

If $P_{n} \rightsquigarrow P_{0}$ and $Q_{n} \leadsto Q_{0}$ , show that $P_{n} \otimes Q_{n} \leadsto P_{0} \otimes Q_{0}$ , even if $Z$ is not separable. (For separable Z, the result follows from (iii); otherwise use (ii).)

Problem 7.10

(i)(ii)

Suppose $\left\{ X_{n}\right\}$ are random elements of $X$ , and $P$ is a probability measure $P$ on $\mathcal{B}(X)$ , for which $\mathbb {P}\left(X_{n} \in B\right) \rightarrow P B$ for every $P$ -continuity set. Let $f$ be a bounded measurable function on $X$ (with no loss of generality assume $0 \leq f \leq 1$ ) that is continuous at all points except those of a $P$ -negligible set $\mathcal{N}$ .

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(i)

For each real $t$ , show that the boundary of the set $\left\{ f \geq t\right\}$ is contained in $\mathcal{N} \cup \left\{ f = t\right\}$ . Deduce that $\left\{ f \geq t\right\}$ is a $P$ -continuity set for almost all (Lebesgue measure) $t$ . Hint: Consider sequences $x_{n} \rightarrow x$ and $y_{n} \rightarrow x$ with $f\left(x_{n}\right) \geq t > f\left(y_{n}\right)$ .

(ii)

Show that $\mathbb {E}\left[f\left(X_{n}\right)\right] = \int_{0}^{1} \mathbb {P}\left(f\left(X_{n}\right) \geq t\right) d t \rightarrow P f$ .

Problem 7.11

Suppose $\mu_{n} \rightarrow \mu$ and $\sigma_{n}^{2} \rightarrow \sigma^{2}$ , with both limits finite. Let $Z$ have a $N(0,1)$ distribution. Show that $\left|\mathbb {E}\left[\ell \left(\mu_{n}+\sigma_{n} Z\right)\right]-\mathbb {E}\left[\ell (\mu +\sigma Z)\right]\right| \leq \left\| \ell \right\|_{B L} \mathbb {E}\left[1 \wedge \left(\left|\mu_{n}-\mu \right|+\left|\sigma_{n}-\sigma \right|\left|Z\right|\right)\right]$ . Deduce, by Dominated Convergence, that $N\left(\mu_{n}, \sigma_{n}^{2}\right) \rightsquigarrow N\left(\mu , \sigma^{2}\right)$ .

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Problem 7.12

(i)(ii)

Suppose $X_{n}$ has a $N\left(\mu_{n}, \sigma_{n}^{2}\right)$ distribution, and $X_{n} \rightsquigarrow P$ .

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(i)

Show that $\mu : = \lim \mu_{n}$ and $\sigma^{2}: = \lim \sigma_{n}^{2}$ must exist as finite limits, and that $P$ must be the $N\left(\mu , \sigma^{2}\right)$ distribution. Hint: Choose $M$ with $P\left\{ M\right\} = 0 = P\left\{ -M\right\}$ and $P[-M, M] > 3 / 4$ . If $\left|\mu_{n}\right| > M$ or if $\sigma_{n}$ is large enough, show that $\mathbb {P}\left(\left|X_{n}\right| > M\right) \geq 1 / 2$ . Show that all convergent subsequences of ( $\mu_{n}, \sigma_{n}$ ) must converge to the same limit.

(ii)

Extend the result to sequences of random vectors. Hint: Use part (i) to prove boundedness of $\left\{ \mu_{n}\right\}$ and each diagonal element of $\left\{ V_{n}\right\}$ . Use Cauchy-Schwarz to deduce that all elements of $\left\{ V_{n}\right\}$ are bounded.

Problem 7.13

Suppose the random variables $\left\{ X_{n}\right\}$ converge in distribution to $X$ , and that $\left\{ A_{n}\right\}$ and $\left\{ B_{n}\right\}$ are sequences of constants with $A_{n} \rightarrow A$ and $B_{n} \rightarrow B$ (both limits finite). Show that $A_{n} X_{n}+B_{n} \rightsquigarrow A X+B$ . Generalize to random vectors.

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Problem 7.14

(i)(ii)(iii)

Let $Y$ be a random $k$ -vector with $\mu : = \mathbb {E}\left[Y\right]$ and $V: = \operatorname {Var}\left[Y\right]$ . Let $V$ have the representation $V = L \Lambda^{2} L^{\prime }$ , with $L$ an orthogonal matrix and $\Lambda : = \operatorname {diag}\left(\lambda_{1}, \ldots , \lambda_{k}\right)$ , each $\lambda_{i}$ nonnegative. Define $R: = L \Lambda$ . Let $W$ be a random $k$ -vector of independent $N(0,1)$ random variables.

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(i)

Show that $\left|\mu \right| \leq \mathbb {E}\left[\left|Y\right|\right]$ . Hint: For a unit vector $u$ in the direction of $\mu$ , use the fact that $u \cdot Y \leq \left|Y\right|$ .

(ii)

Show that $\mathbb {E}\left[\left|R W\right|^{3}\right] = \mathbb {E}\left[\left|\sum_{i} \lambda_{i} W_{i}\right|^{3}\right] \leq (\text{ trace } V)^{3 / 2} \mathbb {E}\left[\left|N(0,1)\right|^{3}\right]$ .

(iii)

Show that $\mathbb {E}\left[\left|\mu +R W\right|^{3}\right] \leq 8 \mathbb {E}\left[\left|Y\right|^{3}\right]+8\mathbb {E}\left[\left|Y\right|^{2}\right]^{3 / 2} \mathbb {E}\left[\left|N(0,1)\right|^{3}\right]$ .

Problem 7.15

(i)(ii)(iii)

Let $f$ be a bounded, measurable, real-valued function on the real line. Let $k$ be a Lebesgue integrable function, with derivative $k^{\prime }$ . Suppose there exists a Lebesgue integrable function $M$ with $\left|k(x+\delta )-k(x)\right| \leq \left|\delta \right| M(x)$ for all $\left|\delta \right| \leq 1$ and all $x$ .

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(i)

Define $g(x): = \int f(x+y) k(y) d y = \int f(z) k(z-x) d z$ . Use Dominated Convergence to justify differentiation under the integral sign to prove that $g$ is differentiable, with derivative $g^{\prime }(x) = -\int f(x+y) k^{\prime }(y) d y$ .

(ii)

Let $k(x): = p(x) \exp \left(-x^{2} / 2\right)$ , with $p$ a polynomial in $x$ . Show that $k$ and each of its derivatives satisfies the stated assumptions. Deduce that the corresponding $g$ has bounded derivatives of all orders. Hint: Consider the case $p(x): = x^{d}$ . Show that $\left|e^{t}-1\right| \leq \left|t\right| e^{\left|t\right|}$ for all real $t$ .

(iii)

For each $\sigma > 0$ , show that the function $x \mapsto k(x / \sigma ) / \sigma$ and each of its derivatives also satisfies the assumptions.

Problem 7.16

Let $\left\{ X_{n}\right\}$ be a sequence of random variables, all defined on the same probability space. If $X_{n} = o_{p}(1)$ , we know from Chapter 2 that there is a subsequence $\left\{ X_{n_{i}}: i \in \mathbb {N}\right\}$ for which $X_{n_{i}}(\omega ) = o(1)$ for almost all $\omega$ . If, instead, $X_{n} = O_{p}(1)$ , must there exist a subsequence for which $X_{n_{i}}(\omega ) = O(1)$ for almost all $\omega$ ?

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Problem 7.17

Let $\psi$ be a strictly increasing function on $\mathbb {R}^{+}$ with $\psi (0) = 0$ and $\psi (t) \rightarrow 1$ as $t \rightarrow \infty$ . Show that a sequence of random vectors $\left\{ X_{n}\right\}$ is of order $O_{p}(1)$ if and only if $\lim \sup \mathbb {E}\left[\psi \left(\left|X_{n}\right|\right)\right] < 1$ .

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Problem 7.18

Let $\left\{ X_{n}\right\}$ and $\left\{ Y_{n}\right\}$ be sequences of random $k$ -vectors, with $X_{n}$ and $Y_{n}$ defined on the same space and independent of each other. Suppose $X_{n}-Y_{n} = O_{p}(1)$ . Show that there exists a sequence of nonrandom vectors $\left\{ a_{n}\right\}$ for which $X_{n}-a_{n} = O_{p}(1)$ .

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Problem 7.19

If $X$ has a Poisson( $\lambda$ ) distribution, show that $\sqrt{X}-\sqrt{\lambda } \leadsto N(0,1 / 4)$ as $\lambda \rightarrow \infty$ .

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Problem 7.20

Let $\left\{ X_{n, i}\right\}$ be a triangular array of random variables, independent within each row and satisfying (a) $\sum_{i} \mathbb {P}\left(\left|X_{n, i}\right| > \epsilon \right) \rightarrow 0$ for each $\epsilon > 0$ , (b) $\sum_{i} \operatorname {Var}\left[X_{n, i} \left\{ \left|X_{n,i}\right| \leq \varepsilon \right\} \right] \to 1$ for each $\epsilon > 0$ .

Show that $\sum_{i} X_{n, i}-A_{n} \leadsto N(0,1)$ , where $A_{n}: = \sum_{i} \mathbb {E}\left[X_{n, i} \left\{ \left|X_{n, i}\right| \leq 1\right\} \right]$ .

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Problem 7.21

(i)(ii)(iii)

Let $\left\{ X_{n, i}\right\}$ be a triangular array of random variables, independent within each row and satisfying

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(i)

$\max_{i}\left|X_{n, i}\right| \rightarrow 0$ in probability,

(ii)

$\sum_{i} \mathbb {E}\left[X_{n, i} \left\{ \left|X_{n, i}\right| \leq \epsilon \right\} \right] \rightarrow \mu$ for each $\epsilon > 0$ ,

(iii)

$\sum_{i} \operatorname {Var}\left[X_{n, i}\left\{ \left|X_{n, i}\right| \leq \epsilon \right\} \right] \rightarrow \sigma^{2} < \infty$ for each $\epsilon > 0$ .

Show that $\sum_{i} X_{n, i} \leadsto N\left(\mu , \sigma^{2}\right)$ .

Problem 7.22

If each of the components $\left\{ X_{n i}\right\}$ , for $i = 1, \ldots , k$ , of a sequence of random $k$ -vectors $\left\{ X_{n}\right\}$ is of order $O_{p}(1)$ , show that $X_{n} = O_{p}(1)$ .

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Problem 7.23

Suppose $f(x) = o(\left|x\right|)$ as $x \rightarrow 0$ , and $g(x) = O\left(\left|x\right|^{k}\right)$ as $x \rightarrow 0$ . Suppose $X_{n} = O_{p}\left(\alpha_{n}\right)$ for some sequence of constants $\alpha_{n}$ tending to zero. Derive a bound for the rate at which $f\left(g\left(X_{n}\right)\right)$ tends to zero.

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