Math Problems Vault

Problem 7.2.1

(a)(b)(c)

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(a)

Suppose $X_{n} \xrightarrow {r} X$ where $r \geq 1$ . Show that $\mathbb {E}\left[\left|X_{n}^{r}\right|\right] \rightarrow \mathbb {E}\left[\left|X^{r}\right|\right]$ .

(b)

Suppose $X_{n} \xrightarrow {1} X$ . Show that $\mathbb {E}\left[X_{n}\right] \rightarrow \mathbb {E}\left[X\right]$ . Is the converse true?

(c)

Suppose $X_{n} \xrightarrow {2} X$ . Show that $\operatorname {Var}\left[X_{n}\right] \rightarrow \operatorname {Var}\left[X\right]$ .

Problem 7.2.2

Suppose $\left|X_{n}\right| \leq Z$ for all $n$ , where $\mathbb {E}\left[Z\right] < \infty$ . Prove that if $X_{n} \xrightarrow {\mathrm{P}} X$ then $X_{n} \xrightarrow {1} X$ .

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Problem 7.2.3

(a)(b)

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(a)

Give a rigorous proof that $\mathbb {E}\left[X Y\right] = \mathbb {E}\left[X\right] \mathbb {E}\left[Y\right]$ for any pair $X, Y$ of independent non-negative random variables on $(\Omega , \mathcal{F}, \mathbb {P})$ with finite means. [Hint: For $k \geq 0, n \geq 1$ , define $X_{n} = k / n$ if $k / n \leq X < (k+1) / n$ , and similarly for $Y_{n}$ . Show that $X_{n}$ and $Y_{n}$ are independent, and $X_{n} \leq X$ , and $Y_{n} \leq Y$ . Deduce that $\mathbb {E}\left[X_{n}\right] \rightarrow \mathbb {E}\left[X\right]$ and $\mathbb {E}\left[Y_{n}\right] \rightarrow \mathbb {E}\left[Y\right]$ , and also $\mathbb {E}\left[X_{n} Y_{n}\right] \rightarrow \mathbb {E}\left[X Y\right]$ .]

(b)

Give an example to show that the product $X Y$ of dependent random variables $X, Y$ with finite means may have $\mathbb {E}\left[X Y\right] = \infty$ .

Problem 7.2.4

Show that convergence in distribution is equivalent to convergence with respect to the Lévy metric of Exercise (7.1.4).

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Problem 7.2.5

(a)(b)

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(a)

Suppose that $X_{n} \xrightarrow {\mathrm{D}} X$ and $Y_{n} \xrightarrow {\mathrm{P}} c$ , where $c$ is a constant. Show that $X_{n} Y_{n} \xrightarrow {\mathrm{D}} c X$ , and that $X_{n} / Y_{n} \xrightarrow {\mathrm{D}} X / c$ if $c \neq 0$ .

(b)

Suppose that $X_{n} \xrightarrow {\mathrm{D}} 0$ and $Y_{n} \xrightarrow {\mathrm{P}} Y$ , and let $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ be continuous. Show that $g\left(X_{n}, Y_{n}\right) \xrightarrow {\mathrm{P}} g(0, Y)$ .

Remark.

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These results are sometimes referred to as 'Slutsky's theorem(s)'.

Problem 7.2.6

Let $X_{1}, X_{2}, \ldots$ be random variables on the probability space $(\Omega , \mathcal{F}, \mathbb {P})$ . Show that the set $A = \left\{ \omega \in \Omega \colon \text{the sequence } X_{n}(\omega ) \text{ converges}\right\}$ is an event (that is, lies in $\mathcal{F}$ ), and that there exists a random variable $X$ (that is, an $\mathcal{F}$ -measurable function $X: \Omega \rightarrow \mathbb {R}$ ) such that $X_{n}(\omega ) \rightarrow X(\omega )$ for $\omega \in A$ .

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Problem 7.2.7

(a)(b)(c)(d)

Let $\left\{ X_{n}\right\}$ be a sequence of random variables, and let $\left\{ c_{n}\right\}$ be a sequence of reals converging to the limit $c$ . Show that the convergence of $X_{n}$ to $X$ entails the convergence of $c_{n} X_{n}$ to $c X$ , for convergence

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(a)

almost surely,

(b)

in $r$ th mean,

(c)

in probability,

(d)

in distribution.

Problem 7.2.8

Let $\left\{ X_{n}\right\}$ be a sequence of independent random variables which converges in probability to the limit $X$ . Show that $X$ is almost surely constant.

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Problem 7.2.9

The sequence of discrete random variables $X_{n}$ , with mass functions $f_{n}$ , is said to converge in total variation to $X$ with mass function $f$ if

\sum _{x}\left|f_{n}(x)-f(x)\right| \rightarrow 0 \quad \text{ as } \quad n \rightarrow \infty

Suppose $X_{n} \rightarrow X$ in total variation, and $u: \mathbb {R} \rightarrow \mathbb {R}$ is bounded. Show that $\mathbb {E}\left[u\left(X_{n}\right)\right] \rightarrow \mathbb {E}\left[u(X)\right]$ .

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Problem 7.2.10

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent Poisson variables with respective parameters $\left\{ \lambda_{r}: r \geq 1\right\}$ . Show that $\sum_{r = 1}^{\infty } X_{r}$ converges or diverges almost surely according as $\sum_{r = 1}^{\infty } \lambda_{r}$ converges or diverges.

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Problem 7.2.11

Let $X_{1}, X_{2}, \ldots$ be independent random variables, uniformly distributed on $\left\{ 1,2, \ldots , m\right\}$ , and let $I_{m} = \min \left\{ r \geq 2: X_{r} = X_{S} \text{ for some } s < r\right\}$ be the earliest index of a coincidence of values. Show that $I_{m} / \sqrt{m}$ converges in distribution as $m \rightarrow \infty$ , with as limit the Rayleigh distribution with density function $f(x) = x e^{-\frac{1}{2} x^{2}}, x > 0$ .

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Problem 7.2.12

(a)(b)

The random variable $X$ has finite moments of all orders, and satisfies $\mathbb {P}\left(X > 0\right) = 1$ , $\mathbb {P}\left(X > x\right) > 0$ for $x > 0$ .

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(a)

Show that $c = \sum_{n = 0}^{\infty } 1 / \mathbb {E}\left[X^{n}\right]$ satisfies $c < \infty$ .

(b)

Let $M$ take values in $\left\{ 1,2, \ldots \right\}$ with mass function $f(m) = 1 /\left(c \mathbb {E}\left[X^{m}\right]\right)$ for $m \geq 1$ . Show that $\mathbb {E}\left[x^{M}\right] < \infty$ for all $x \geq 0$ , while $\mathbb {E}\left[X^{M}\right] = \infty$ .