Math Problems Vault

Problem C.1

Let $f$ be the convex function, taking values in $\mathbb {R} \cup \left\{ \infty \right\}$ , defined by

f(x, y) = \begin{cases} -y^{1 / 2} & \text{ for } 0 \leq 1 \text{ and } x \in \mathbb {R} \\ \infty & \text{ otherwise } \end{cases}

Let $T_{0}$ denote the linear function defined on the $x$ -axis by $T_{0}(x, 0): = 0$ for all $x \in \mathbb {R}$ . Show that $T_{0}$ has no extension to a linear functional on $\mathbb {R}^{2}$ for which $T(x, y) \leq f(x, y)$ everywhere, even though $T_{0} \leq f$ along the $x$ -axis.

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Problem C.2

(i)(ii)(iii)

Suppose $X$ is a random variable for which the moment generating function, $M(t): = \mathbb {E}\left[\exp (t X)\right]$ , exists (and is finite) for $t$ in an open interval $J$ about the origin of the real line. Write $\mathbb {P}_{t}$ for the probability measure with density $e^{t X} / M(t)$ with respect to $\mathbb {P}$ , for $t \in J$ , with corresponding variance $\operatorname {var}_{t}(\cdot )$ . Define $\Lambda (t): = \log M(t)$ .

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(i)

Use Dominated Convergence to justify the operations needed to show that

\begin{aligned} \Lambda ^{\prime }(t) & = M^{\prime }(t) / M(t) = \mathbb {E}\left[X e^{t X} / M(t)\right] = \mathbb {E}_{t}\left[X\right] \\ \Lambda ^{\prime \prime }(t) & = \left(M(t) M^{\prime \prime }(t)-M^{\prime }(t)^{2}\right) / M(t)^{2} = \operatorname {var}_{t}(X) \end{aligned}

(ii)

Deduce that $\Lambda$ is a convex function on $J$ .

(iii)

Show that $\Lambda$ achieves its minimum at $t = 0$ if $\mathbb {E}\left[X\right] = 0$ .

Problem C.3

(i)(ii)(iii)

Let $Q$ be a probability measure defined on a finite interval [a,b]. Write $\sigma_{Q}^{2}$ for its variance.

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(i)

Show that $\sigma_{Q}^{2} \leq (b-a)^{2} / 4$ . Hint: Reduce to the case $b = -a$ , noting that $\sigma_{Q}^{2} \leq Q^{x}\left(x^{2}\right)$ .

(ii)

Suppose also that $Q^{x}(x) = 0$ . Define $\Lambda (t): = \log \left(Q^{x} e^{x t}\right)$ , for $t \in \mathbb {R}$ . Show that $\Lambda^{\prime \prime }(t) \leq (b-a)^{2} / 4$ , and hence $\Lambda (t) \leq t^{2}(b-a)^{2} / 8$ for all $t \in \mathbb {R}$ .

(iii)

(Hoeffding 1963) Let $X_{1}, \ldots , X_{n}$ be independent random, variables with zero expected values, and with $X_{i}$ taking values only in a finite interval $\left[a_{i}, b_{i}\right]$ . For $\epsilon > 0$ , show that

\mathbb {P}\left(X_{1}+\ldots +X_{n} \geq \epsilon \right) \leq \inf _{t > 0} e^{-\epsilon t} \prod _{i} \mathbb {E}\left[e^{t X_{i}}\right] \leq \exp \left(-2 \epsilon ^{2} / \sum _{i}\left(b_{i}-a_{i}\right)^{2}\right)

Problem C.4

(i)(ii)

Let $P$ be a probability measure on $\mathbb {R}^{k}$ . Define $M(t) ; = P^{x}\left(e^{x \cdot t}\right)$ for $t \in \mathbb {R}^{k}$ .

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(i)

Show that the set $C: = \left\{ t \in \mathbb {R}^{k}: M(t) < \infty \right\}$ is convex.

(ii)

Show that $\log M(t)$ is convex on rel-int $(C)$ .

Problem C.5

Let $f$ be a convex increasing function on $\mathbb {R}^{+}$ . Show that there exists an increasing sequence of convex, increasing functions $f_{n}$ , with each $f_{n}^{\prime \prime }$ bounded and continuous, such that $0 \leq f_{n}(x) \leq f_{n+1}(x) \uparrow f(x)$ for each $x$ .

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