Brownian bridge and empirical processes

Let $B_{0}(t) \equiv B(t)-t B(1), 0 \leq t \leq 1$ (where $\left\{ B(t): t \geq 0\right\}$ is SBM) be the Brownian bridge. Find the distribution of $X(t) \equiv (1+t) B_{0}\left(\frac{t}{1+t}\right)$, $t \geq 0$.

Let $X$ be a Brownian bridge. (This will be studied further in Section 35.2.) This means that $X$ is a mean zero Gaussian process with

Identify the reproducing kernel Hilbert space for $X$.

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.

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.

Prove that the Brownian bridge (see the definition in $\S 9$ of Chapter V) is an inhomogeneous Markov process.

Definition from $\S 9$ of Chapter V: $W^{0}$ is the Brownian bridge, defined as the process $W^{0}(t) = \left\{ W(t)-t W(1), t \in [0,1]\right\}$, where $W = \left\{ W(t), t \in [0,1]\right\}$ is a Wiener process on the interval $[0,1]$.

Using the Itô formula, prove that the solution of the (one-dimensional) equation

is the process

Let $X_{t}$ be the solution of the (one-dimensional) equation

(continuation of Exercise 23). Prove that $X_{t} \rightarrow \beta$ a.s. as $t \rightarrow T-$. Thus, the solution of equation (109) is a Brownian bridge over the interval $[0, T]$ with pinned endpoints $X_{0} = \alpha$ and $X_{T} = \beta$. The standard Brownian bridge is obtained for $T = 1$ and $\alpha = \beta = 0$.

Show that the autocovariance function of the Brownian bridge is $c(s, t) = \min \left\{ s, t\right\} -s t, 0 \leq s, t \leq 1$.

Let $W$ be a standard Wiener process, and let $\widehat{W}(t) = W(t)-t W(1)$. Show that $\left\{ \widehat{W}(t): 0 \leq t \leq 1\right\}$ is a Brownian bridge.

If $W$ is a Wiener process with $W(0) = 0$, show that $\widetilde{W}(t) = (1-t) W(t /(1-t))$ for $0 \leq t < 1$, $\widetilde{W}(1) = 0$, defines a Brownian bridge.

Deduce, with help from Corollary (13.4.14), that the probability of the Brownian bridge ever rising above the height $m > 0$ is $e^{-2 m^{2}}$.

Let $0 < s < t < 1$. Show that the probability that the Brownian bridge has no zeros in the interval $(s, t)$ is $(2 / \pi ) \cos^{-1} \sqrt{(t-s) /[t(1-s)]}$.

Let $W$ be a standard Wiener process. Find the conditional joint density function of $W\left(t_{2}\right)$ and $W\left(t_{3}\right)$ given that $W\left(t_{1}\right) = W\left(t_{4}\right) = 0$, where $t_{1} < t_{2} < t_{3} < t_{4}$.

Show that the conditional correlation of $W\left(t_{2}\right)$ and $W\left(t_{3}\right)$ is

Let $U_{1}, U_{2}, \ldots$ be independent random variables with the uniform distribution on $[0,1]$. Let $I_{j}(x)$ be the indicator function of the event $\left\{ U_{j} \leq x\right\}$, and define

The function $F_{n}$ is called the 'empirical distribution function' of the $U_{j}$.

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = w(t) - tw(1)$, considered on the time interval $t \in [0, 1]$.

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = (1-t)w(t/(1-t))$, $t \in [0, 1)$, $w^{(0)}(1) = 0$.

Let $\left(B_{t}, t \in [0,1]\right)$ be a Gaussian process with zero mean function and covariance function $r(s, t)=\min (t_1, t_2)-t_1 t_2$.

Let $w_{t}$ be a Wiener process. Find the joint density of the values $w_{u}$ and $w_{v}$, $0 < u < v < 1$, given that $w_{1} = 0$.

Let $w_{t}$ be a Wiener process. Find the covariance of the values $w_{s}$ and $w_{t}, s < t < 1$, given that $w_{1} = 0$.

Let $w_{t}$ be a Wiener process. Find the correlation function of the process $w_{t}^{(0)} = w_{t}-t w_{1}$, considered on the segment $0 \leqslant t \leqslant 1$ (the conditional Wiener process).

Let $w_{t}^{(0)}, \quad 0 \leqslant t \leqslant 1$, be the conditional Wiener process defined in the previous problem. Prove that the process $\bar{w}_{t} = (1 + t) w_{1 /(1+t)}^{(0)}, \quad t \geqslant 0,$ is a Wiener process.

Let $w_{t}$ be a Wiener process. Find the conditional density of $w_{t}, t_{1} < t < t_{2}$, given that

Let $w_{t}$ be a Wiener process. Find the distribution function and density of the random variable $\max_{0 \leqslant s \leqslant 1} w_{s}$, given that $w_{1} = 0$.

Let $w_{t}$ be a Wiener process. Find the distribution function of the random variable $\sup_{0 \leqslant t \leqslant 1}\left|w_{t}\right|$, given that $w_{1} = 0$.

Let $(B(t))_{t \geqslant 0}$ be a $B M^{1}$.

Let $B = \left(B_{t}\right)_{t \in [0,1]}$ be a $\mathrm{BM}^{1}$ and set $W = \left(W_{t}\right)_{t \in [0,1]}$ where $W_{t}: = B_{t} - t B_{1}$.