Math Problems Vault

Problem 31.1

Suppose $f$ is bounded, $X_{n}$ converges to $X$ weakly, and also that $\mathbb {P}\left(X \in D_{f}\right)=0$ , where $D_{f}=\left\{ x: f \text{ is not continuous at } x\right\}$ . Show that $f\left(X_{n}\right)$ converges weakly to $f(X)$ .

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Problem 31.2

Suppose a sequence $\left\{ X_{n}\right\}$ is uniformly integrable and $X_{n}$ converges to $X$ weakly. Show $\mathbb {E}\left[X_{n}\right] \rightarrow \mathbb {E}\left[X\right]$ .

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Problem 31.3

Give an example of a sequence of random variables $X_{n}$ converging weakly to $X$ and where each $X_{n}$ is integrable, but $X$ is not integrable.

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Problem 31.4

Suppose $X_{n}$ converges weakly to $X$ and each $X_{n}$ is non-negative. Prove that

\mathbb {E}\left[X\right] \leq \liminf _{n \rightarrow \infty } \mathbb {E}\left[X_{n}\right]

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Problem 31.5

Suppose $X_{n}$ converges weakly to $X$ and each $X_{n}$ has the property that with probability one,

\left|X_{n}(t)-X_{n}(s)\right| \leq |t-s|, \quad s, t \leq 1

(This might arise, for example, if each $X_{n}$ is of the form $X_{n}(t)=\int_{0}^{t} Y_{n}(s) d s$ and each $Y_{n}$ is bounded by 1.) Prove that $X$ has this same property, that is, with probability one,

|X(t)-X(s)| \leq |t-s|, \quad s, t \leq 1

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Problem 31.6

(1)(2)

Here is a way to prove one direction of Lebesgue's theorem on Riemann integrable functions.

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(1)

For each $n \geq 1$ and each $i \leq n$ , let $x_{i n}$ be a point in $[(i-1) / n, i / n)$ . Let $\mathbb {P}_{n}$ be the probability measure that assigns mass $1 / n$ to each point $x_{i n}, i=1,2, \ldots , n$ . Show that $\mathbb {P}_{n}$ converges weakly to $\mathbb {P}$ , where $\mathbb {P}$ is a Lebesgue measure on $[0,1]$ .

(2)

Suppose $f$ is a bounded function which is continuous at almost every point of $[0,1]$ . Show that $\int f d \mathbb {P}_{n} \rightarrow \int f d \mathbb {P}$ . Note that $\int f d \mathbb {P}_{n}$ is a Riemann sum approximation to $\int_{0}^{1} f(x) d x$ .