Math Problems Vault

Problem 28.1

Show that $\mu_{n} \rightarrow_{\iota } \mu$ implies $\mu \left(R^{1}\right) \leq \liminf_{n} \mu_{n}\left(R^{1}\right)$ . Thus in vague convergence mass can "escape to infinity" but mass cannot "enter from infinity."

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Problem 28.2

(a)(b)

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(a)

Show that $\mu_{n} \rightarrow_{\iota } \mu$ if and only if (28.2) holds for every continuous $f$ with bounded support.

(b)

Show that if $\mu_{n} \rightarrow_{\iota } \mu$ but (28.1) does not hold, then there is a continuous $f$ vanishing at $\pm \infty$ for which (28.2) does not hold.

Problem 28.3

(a)(b)(c)

$23.7 \uparrow$ Suppose that $N, Y_{1}, Y_{2}, \ldots$ are independent, the $Y_{n}$ have a common distribution function $F$ , and $N$ has the Poisson distribution with mean $\alpha$ . Then $S=Y_{1}+\cdots +Y_{N}$ has the compound Poisson distribution.

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(a)

Show that the distribution of $S$ is infinitely divisible. Note that $S$ may not have a mean.

(b)

The distribution function of $S$ is $\Sigma_{n=0}^{\infty } e^{-\alpha } \alpha^{n} F^{n *}(x) / n!$ , where $F^{n *}$ is the $n$ -fold convolution of $F$ (a unit jump at 0 for $n=0$ ). The characteristic function of $S$ is $\exp \alpha \int_{-\infty }^{\infty }\left(e^{i t x}-1\right) d F(x)$ .

(c)

Show that, if $F$ has mean 0 and finite variance, then the canonical measure $\mu$ in (28.6) is specified by $\mu (A)=\alpha \int_{A} x^{2} d F(x)$ .

Problem 28.4

(a)(b)

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(a)

Let $\nu$ be a finite measure, and define

\varphi (t)=\exp \left[i \gamma t+\int _{-\infty }^{\infty }\left(e^{i t x}-1-\frac{i t x}{1+x^{2}}\right) \frac{1+x^{2}}{x^{2}} \nu (d x)\right], \tag {28.12}

where the integrand is $-t^{2} / 2$ at the origin. Show that this is the characteristic function of an infinitely divisible distribution.

(b)

Show that the Cauchy distribution (see the table on p. 348) is the case where $\gamma =0$ and $\nu$ has density $\pi^{-1}\left(1+x^{2}\right)^{-1}$ with respect to Lebesgue measure.

Problem 28.5

Show that the Cauchy, exponential, and gamma (see (20.47)) distributions are infinitely divisible.

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Problem 28.6

(a)(b)(c)

Find the canonical representation (28.6) of the exponential distribution with mean 1:

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(a)

The characteristic function is $\int_{0}^{\infty } e^{i t x} e^{-x} d x=(1-i t)^{-1}=\varphi (t)$ .

(b)

Show that (use the principal branch of the logarithm or else operate formally for the moment) $d(\log \varphi (t)) / d t=i \varphi (t)=i \int_{0}^{\infty } e^{i t x} e^{-x} d x$ . Integrate with respect to $t$ to obtain

\frac{1}{1-i t}=\exp \int _{0}^{\infty }\left(e^{i t x}-1\right) \frac{e^{-x}}{x} d x \tag {28.13}

Verify (28.13) after the fact by showing that the ratio of the two sides has derivative 0.

(c)

Multiply (28.13) by $e^{-i t}$ to center the exponential distribution at its mean: The canonical measure $\mu$ has density $x e^{-x}$ over $(0, \infty )$ .

Problem 28.7

$\uparrow$ If $X$ and $Y$ are independent and each has the exponential density $e^{-x}$ , then $X-Y$ has the double exponential density $\frac{1}{2} e^{-\left|x\right|}$ (see the table on p. 348). Show that its characteristic function is

\frac{1}{1+t^{2}}=\exp \int _{-\infty }^{\infty }\left(e^{i t x}-1-i t x\right) \frac{1}{x^{2}}\left|x\right| e^{-\left|x\right|} d x

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Problem 28.8

$\uparrow$ Suppose $X_{1}, X_{2}, \ldots$ are independent and each has the double exponential density. Show that $\sum_{n=1}^{\infty } X_{n} / n$ converges with probability 1. Show that the distribution of the sum is infinitely divisible and that its canonical measure has density $|x| e^{-|x|} /\left(1-e^{-|x|}\right)=\sum_{n=1}^{\infty }|x| e^{-|n x|}$ .

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Problem 28.9

$26.8 \uparrow$ Show that for the gamma density $e^{-x} x^{u-1} / \Gamma (u)$ the canonical measure has density $u x e^{-x}$ over $(0, \infty )$ .

The remaining problems require the notion of a stable law. A distribution function $F$ is stable if for each $n$ there exist constants $a_{n}$ and $b_{n}, a_{n}>0$ , such that, if $X_{1}, \ldots , X_{n}$ are independent and have distribution function $F$ , then $a_{n}^{-1}\left(X_{1}+\cdots +X_{n}\right)+b_{n}$ also has distribution function $F$ .

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Problem 28.10

Suppose that for all $a, a^{\prime }, b, b^{\prime }$ there exist $a^{\prime \prime }, b^{\prime \prime }$ (here $a, a^{\prime }, a^{\prime \prime }$ are all positive) such that $F(a x+b) * F\left(a^{\prime } x+b^{\prime }\right)=F\left(a^{\prime \prime } x+b^{\prime \prime }\right)$ . Show that $F$ is stable.

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Problem 28.11

Show that a stable law is infinitely divisible.

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Problem 28.12

Show that the Poisson law, although infinitely divisible, is not stable.

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Problem 28.13

Show that the normal and Cauchy laws are stable.

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Problem 28.14

$28.10 \uparrow$ Suppose that $F$ has mean 0 and variance 1 and that the dependence of $a^{\prime \prime }, b^{\prime \prime }$ on $a, a^{\prime }, b, b^{\prime }$ is such that

F\left(\frac{x}{\sigma _{1}}\right) * F\left(\frac{x}{\sigma _{2}}\right)=F\left(\frac{x}{\sqrt{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)

Show that $F$ is the standard normal distribution.

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Problem 28.15

(a)(b)

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(a)

Let $Y_{n k}$ be independent random variables having the Poisson distribution with mean $c n^{\alpha } /|k|^{1+\alpha }$ , where $c>0$ and $0<\alpha <2$ . Let $Z_{n}=n^{-1} \sum_{k=-n^{2}}^{n^{2}} k Y_{n k}$ (omit $k=0$ in the sum), and show that if $c$ is properly chosen then the characteristic function of $Z_{n}$ converges to $e^{-\left|t\right|^{\alpha }}$ .

(b)

Show for $0<\alpha \leq 2$ that $e^{-\left|t\right|^{\alpha }}$ is the characteristic function of a symmetric stable distribution; it is called the symmetric stable law of exponent $\alpha$ . The case $\alpha =2$ is the normal law, and $\alpha =1$ is the Cauchy law.