Applications of weak convergence

Prove Lemma 35.8.

Lemma 35.8: Suppose $S_{n}$ is a binomial random variable with parameters $n$ and $p$. Then there exists a constant $c$ not depending on $n$ or $p$ such that

and

Prove that the finite-dimensional distributions of $Z_{n}$ in Theorem 35.10 converge to those of a Brownian bridge.

Setup: let $X_{i}, i=1, \ldots , n$, be i.i.d. random variables uniformly distributed on $[0,1]$, let $F_{n}(t)=\frac{1}{n} \sum_{i=1}^{n} \; \mathbb {1}_{[0, t]}\left(X_{i}\right)$ be the empirical process, and let

Theorem 35.10: The $Z_{n}$ converge weakly to a Brownian bridge with respect to the topology of $D[0,1]$.

If $W_{t}^{0}$ is a Brownian bridge, prove that $Y_{t}=W_{1-t}^{0}$ is also a Brownian bridge.

Let $t_{0}<1$. The SDE (35.5) has a unique solution when $X_{0}=0$ is replaced by $X_{0}=x$. Let $\mathbb {P}[x]$ be the law of the solution when $X_{0}=x$ and let $Z_{t}$ be the canonical process. Show that $\left(Z_{t}, \mathbb {P}[x]\right)$ is not a Markov process.

Let $N_{t}(A)$ be a Poisson point process with respect to the measure space $(\mathcal{S}, m)$ and let $A_{s}, s>0$, be an increasing sequence of subsets of $\mathcal{S}$ with $m\left(A_{s}\right) \rightarrow \infty$ as $s \rightarrow \infty$. Does

converge weakly with respect to $D[0,1]$ as $s \rightarrow \infty$ ? What is the limit?

This can be applied to get central limit theorems for the number of downcrossings of a Brownian motion, for example.

This exercise asks you to prove that the Poisson process conditioned to be equal to $n$ at time 1 has the same law as $n$ times the empirical process. Here is the precise statement. Suppose $P_{t}$ is a Poisson process with parameter $\lambda >0$. Let $\mathbb {Q}$ be the law of $\left\{ P_{t}, t \in [0,1]\right\}$ conditioned so that $P_{1}=n$. Thus $\mathbb {Q}$ is a probability on $D[0,1]$ with

Since $\left(P_{1}=n\right)$ is an event with positive probability, there is no difficulty defining these conditional probabilities. Prove that $\mathbb {Q}$ is also the law of $\left\{ n F_{n}(t), t \in [0,1]\right\}$, where $F_{n}$ is defined in Section 35.3.