Math Problems Vault

Problem 5.9.1

Let $X_{n}$ be a discrete random variable taking values in $\left\{ 1,2, \ldots , n\right\}$ , each possible value having probability $n^{-1}$ . Show that, as $n \rightarrow \infty , \mathbb {P}\left(n^{-1} X_{n} \leq y\right) \rightarrow y$ , for $0 \leq y \leq 1$ .

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Problem 5.9.2

(a)(b)

Let $X_{n}$ have distribution function

F_{n}(x) = x-\frac{\sin (2 n \pi x)}{2 n \pi }, \quad 0 \leq x \leq 1

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(a)

Show that $F_{n}$ is indeed a distribution function, and that $X_{n}$ has a density function.

(b)

Show that, as $n \rightarrow \infty , F_{n}$ converges to the uniform distribution function, but that the density function of $F_{n}$ does not converge to the uniform density function.

Problem 5.9.3

A coin is tossed repeatedly, with heads turning up with probability $p$ on each toss. Let $N$ be the minimum number of tosses required to obtain $k$ heads. Show that, as $p \downarrow 0$ , the distribution function of $2 N p$ converges to that of a gamma distribution.

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Problem 5.9.4

If $X$ is an integer-valued random variable with characteristic function $\phi$ , show that

\mathbb {P}\left(X = k\right) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } e^{-i t k} \phi (t) d t

What is the corresponding result for a random variable whose distribution is arithmetic with span $\lambda$ (that is, there is probability one that $X$ is a multiple of $\lambda$ , and $\lambda$ is the largest positive number with this property)?

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Problem 5.9.5

Use the inversion theorem to show that

\int _{-\infty }^{\infty } \frac{\sin (a t) \sin (b t)}{t^{2}} d t = \pi \min \left\{ a, b\right\}

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Problem 5.9.6

Let $f_{n}(x)$ be a differentiable function on $\mathbb {R}$ with a a global maximum at $a > 0$ , and such that $\int_{0}^{\infty } \exp \left\{ f_{n}(x)\right\} d x < \infty$ . Laplace's method of steepest descent (related to Watson's lemma and saddlepoint methods) asserts under mild conditions that

\int _{0}^{\infty } \exp \left\{ f_{n}(x)\right\} d x \sim \int _{0}^{\infty } \exp \left\{ f_{n}(a)+\frac{1}{2}(x-a)^{2} f_{n}^{\prime \prime }(a)\right\} d x \quad \text{ as } n \rightarrow \infty

By setting $f_{n}(x) = n \log x-x$ , prove Stirling's formula: $n!\sim n^{n} e^{-n} \sqrt{2 \pi n}$ .

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Problem 5.9.7

Let $\mathbf{X} = \left(X_{1}, X_{2}, \ldots , X_{n}\right)$ have the multivariate normal distribution with zero means, and covariance matrix $\mathbf{V} = \left(v_{i j}\right)$ satisfying $\left|\mathbf{V}\right| > 0$ and $v_{i j} > 0$ for all $i, j$ . Show that

\frac{\partial f}{\partial v_{i j}} = \begin{cases} \frac{\partial ^{2} f}{\partial x_{i} \partial x_{j}} & \text{ if } i \neq j \\ \frac{1}{2} \frac{\partial ^{2} f}{\partial x_{i}^{2}} & \text{ if } i = j\end{cases}

and deduce that $\mathbb {P}\left(\max_{k \leq n} X_{k} \leq u\right) \geq \prod_{k = 1}^{n} \mathbb {P}\left(X_{k} \leq u\right)$ .

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Problem 5.9.8

Let $X_{1}, X_{2}$ have a bivariate normal distribution with zero means, unit variances, and correlation $\rho$ . Use the inversion theorem to show that

\frac{\partial }{\partial \rho } \mathbb {P}\left(X_{1} > 0, X_{2} > 0\right) = \frac{1}{2 \pi \sqrt{1-\rho ^{2}}}

Hence find $\mathbb {P}\left(X_{1} > 0, X_{2} > 0\right)$ .

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Problem 5.9.9

(a)(b)

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(a)

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed random variables with characteristic function satisfying $\phi (t) = 1-c\left|t\right|+\mathrm{o}(t)$ as $t \rightarrow 0$ , where $c > 0$ . Show that, as $n \rightarrow \infty$ , the distribution of $Y_{n} = \left(X_{1}+X_{2}+\cdots +X_{n}\right) /(c n)$ converges to the Cauchy distribution.

(b)

Let $U$ have the uniform distribution on $[-1,1]$ , and show that

\phi _{1 / U}(t) = 1-\left|t\right| \int _{\left|t\right|}^{\infty } \frac{1-\cos x}{x^{2}} d x

When $U_{1}, U_{2}, \ldots , U_{n}$ are independent and distributed as $U$ , write down the limiting distribution as $n \rightarrow \infty$ of $Y_{n} = (2 /(n \pi )) \sum_{r = 1}^{n} U_{r}^{-1}$ .