Math Problems Vault

Problem 8.1

(i)(ii)

Let $f = \left(f_{1}, \ldots , f_{k}\right)$ be a vector of $\mu$ -integrable, real-valued functions. Define $\mu f$ as the vector $\left(\mu f_{1}, \ldots , \mu f_{k}\right)$ .

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(i)

Show that $\left|\mu f\right| \leq \mu \left|f\right|$ , where $\left|\cdot \right|$ denotes the usual Euclidean norm. Hint: Let $\alpha$ be a unit vector in the direction of $\mu f$ . Note that $\alpha^{\prime } f \leq \left|f\right|$ .

(ii)

Let $f = f_{1}+i f_{2}$ be a complex-valued function, with $\mu$ -integrable real and imaginary parts $f_{1}$ and $f_{2}$ . Show that $\left|\mu f\right| \leq \mu \left|f\right|$ , where $\left|\cdot \right|$ denotes the modulus of a complex number.

Problem 8.2

Suppose a Fourier transform has $\left|\psi_{P}\left(t_{0}\right)\right| = 1$ for some nonzero $t_{0}$ . That is, $\psi_{P}\left(t_{0}\right) = \exp \left(i \theta_{0}\right)$ for some real $\theta_{0}$ . Show that $P$ concentrates on the lattice of points $\left\{ \left(\theta_{0}+2 n \pi \right) / t_{0}: n \in \mathbb {Z}\right\}$ .

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Problem 8.3

(i)(ii)(iii)

Suppose is $f$ is both integrable with respect to Lebesgue measure $\mathfrak {m}$ on the real line and absolutely continuous, in the sense that $f(x) = \mathfrak {m}^{x}(\left\{ t \leq x\right\} \dot{f}(t))$ , for all $x$ , for some integrable function $\dot{f}$ .

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(i)

Show that $f(x) \rightarrow 0$ as $\left|x\right| \rightarrow \infty$ . Hint: Show $\mathfrak {m}^{t} \dot{f}(t) = 0$ to handle $x \rightarrow \infty$ .

(ii)

Show that $\mathfrak {m}^{x}\left(f(x) e^{i x t}\right) = -\mathfrak {m}^{x}\left(e^{i x t} \dot{f}(x)\right) /(i t)$ for $t \neq 0$ . Hint: For safe Fubini, write the left-hand side as $\lim_{C \rightarrow \infty } \mathfrak {m}^{x}\left(e^{i x t}\left\{ \left|x\right| \leq C\right\} \mathfrak {m}^{s}(\dot{f}(s)\left\{ s \leq x\right\} )\right)$ .

(iii)

If a probability density has Lebesgue integrable derivatives up to $k$ th order, prove that its Fourier transform is of order $O\left(\left|t\right|^{-k}\right)$ as $\left|t\right| \rightarrow \infty$ .

Problem 8.4

Let $\mathfrak {m}$ denote Lebesgue measure on $\mathcal{B}(\mathbb {R})$ . For each $f$ in $\mathcal{L}^{1}(\mathfrak {m})$ show that $\mathrm{m}^{x}\left(f(x) e^{i x t}\right) \rightarrow 0$ as $t \rightarrow \pm \infty$ .

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Problem 8.5

(i)(ii)

Let $g(\cdot )$ be a bounded real function on the real line with bounded derivatives up to order $k+1$ , and let $X$ be a random variable for which $\mathbb {E}\left[\left|X\right|^{k}\right] < \infty$ . Let $R_{k}(\cdot )$ be the remainder in the Taylor expansion up to $k$ th power:

g(x) = g(0)+x g^{\prime }(0)+\ldots +\frac{x^{k}}{k!} g^{(k)}(0)+R_{k}(x)

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(i)

Show that $\left|R_{k}(x)\right| \leq C \min \left(\left|x\right|^{k},\left|x\right|^{k+1}\right)$ , for some constant $C$ .

(ii)

Invoke Dominated Convergence to show that

\mathbb {E}\left[g(X t)\right] = g(0)+t g^{\prime }(0) \mathbb {E}\left[X\right]+\ldots +\frac{t^{k}}{k!} g^{(k)}(0) \mathbb {E}\left[X^{k}\right]+o\left(t^{k}\right) \quad \text{ as } t \rightarrow 0

Problem 8.6

(i)(ii)(iii)

Let $P$ denote the double-exponential distribution, given by the density $p(x) = 1 / 2 \exp (-\left|x\right|)$ with respect to Lebesgue measure on $\mathcal{B}(\mathbb {R})$ .

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(i)

Show that $P^{x} e^{x s} = 1 /\left(1-s^{2}\right)$ for real $s$ with $\left|s\right| < 1$ .

(ii)

By a leap of faith (or by an appeal to analytic continuation), deduce that $\psi_{P}(t) = 1 /\left(1+t^{2}\right)$ for $t \in \mathbb {R}$ .

(iii)

The Cauchy probability distribution is given by the density $q(x) = \pi^{-1} /\left(1+x^{2}\right)$ . Apply the inversion formula from Theorem $\langle 10 \rangle$ to deduce that the Cauchy distribution has Fourier transform $\exp (-\left|t\right|)$ .

Remark.

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Theorem 10: If $P$ has Fourier transform $\psi$ for which $\int_{-\infty }^{\infty }\left|\psi (z)\right| d z < \infty$ , then $P$ is absolutely continuous with respect to Lebesgue measure, with a density $f(y): = \frac{1}{2 \pi } \int_{-\infty }^{\infty } \psi (z) \exp (-i z y) d z$ that is bounded and uniformly continuous.

Problem 8.7

Suppose $(X, Y)$ has a multivariate normal distribution with $\operatorname {cov}(X, Y) = 0$ . Show that $X$ and $Y$ are independent.

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Problem 8.8

Let $X_{1}, \ldots , X_{n}$ be independent, Uniform $(0,1)$ distributed random variables. Calculate the logarithm of the Fourier transform of $n^{-1 / 2}\left(X_{1}+\ldots +X_{n}-\frac{1}{2} n\right)$ up to a remainder term of order $O\left(n^{-2}\right)$ .

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Problem 8.9

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

Suppose $X$ is a random variable with Fourier transform $\psi_{X}$ .

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(i)

For each $\delta > 0$ , show that

\frac{1}{\delta } \int _{0}^{\delta }\left(1-\Re \psi _{X}(t)\right) d t = \mathbb {E}\left[1-\frac{\sin X \delta }{X \delta }\right] \geq c_{0} \mathbb {P}\left(\left|X\right| \geq 1 / \delta \right)

where $1-c_{0}: = \sup_{\left|x\right| \geq 1}\left|\sin x / x\right|$ .

(ii)

If $\psi$ is a complex-valued function, continuous at the origin, show that

\frac{1}{\delta } \int _{0}^{\delta } \Re (\psi (0)-\psi (t)) d t \rightarrow 0 \quad \text{ as } \delta \rightarrow 0

(iii)

Suppose $\left\{ X_{n}\right\}$ is a sequence of random variables whose Fourier transforms $\psi_{n}$ converge pointwise to a function $\psi$ that is continuous at the origin. Show that $X_{n}$ is of order $O_{p}(1)$ .

(iv)

Show that the $\psi$ from part (iii) equals the Fourier transform of some random variable $Z$ representing the limit in distribution of a subsequence of $\left\{ X_{n}\right\}$ . Deduce via Theorem $\langle 12 \rangle$ that $X_{n} \rightsquigarrow Z$ .

Remark.

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Theorem 12 (Continuity Theorem): Let $X, X_{1}, X_{2}, \ldots$ be random variables with Fourier transforms $\psi , \psi_{1}, \psi_{2}, \ldots$ for which $\psi_{n}(t) \rightarrow \psi (t)$ for each real $t$ . Then $X_{n} \rightsquigarrow X$ .

Problem 8.10

(i)(ii)

Suppose a random variable $X$ has a Fourier transform $\psi$ for which $\psi (t) = 1+O\left(t^{2}\right)$ near $t = 0$ .

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(i)

Show that there exists a finite constant $C$ such that

C \geq \frac{\Re (1-\psi (t))}{t^{2}} \geq \mathbb {E}\left[\frac{1-\cos (X t)}{(X t)^{2}} X^{2}\left\{ 0 < \left|X\right| \leq M\right\} \right]

for all $M$ . Deduce via Dominated Convergence that $2 C \geq \mathbb {E}\left[X^{2}\right]$ .

(ii)

Show that $\mathbb {E}\left[X\right] = 0$ .

Problem 8.11

Suppose $X$ is a random variable, and $f(z, X)$ is a jointly measurable function that is analytic in a neighborhood $\mathcal{N}: = \left\{ z \in \mathbb {C}:\left|z-z_{0}\right| < \delta \right\}$ of a point $z_{0}$ in the complex plane. Write $f^{\prime }(z, X)$ for the derivative with respect to $z$ . Suppose $\left|f^{\prime }(z, X)\right| \leq M(X)$ for $\left|z-z_{0}\right| \leq \delta$ , where $\mathbb {E}\left[M(X)\right] < \infty$ . Show that $\mathbb {E}\left[f(z, X)\right]$ is analytic in $\mathcal{N}$ , with derivative $\mathbb {E}\left[f^{\prime }(z, X)\right]$ .

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Problem 8.12

(i)(ii)

Suppose a random variable $X$ has Fourier transform $\phi (t) = \exp \left(i c t-d t^{2}\right)$ , for complex numbers $c: = c_{1}+i c_{2}$ and $d: = d_{1}+i d_{2}$ .

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(i)

Deduce from the facts that $\left|\phi (t)\right| \leq 1$ and $\phi (t) = 1-c_{2} t+i c_{1} t+o(t)$ near $t = 0$ that $c_{2} = 0$ .

(ii)

Show that $X-c_{1}$ has Fourier transform $\exp \left(-d t^{2}\right) = 1+O\left(t^{2}\right)$ near $t = 0$ . Deduce from Theorem $\langle 2 \rangle$ and Problem [10] that $d = \mathbb {E}\left[\left|X-c_{1}\right|^{2}\right]$ , which is nonnegative.

Remark.

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Theorem 2: If $\mathbb {E}\left[\left|X\right|^{k}\right] < \infty$ , for a positive integer $k$ , then the Fourier transform has the approximation $\psi_{X}(t) = 1+i t \mathbb {E}\left[X\right]+\frac{(i t)^{2}}{2 !} \mathbb {E}\left[X^{2}\right]+\ldots +\frac{(i t)^{k}}{k !} \mathbb {E}\left[X^{k}\right]+o\left(t^{k}\right)$ for $t$ near 0.