Math Problems Vault

Problem 33.1

Finish the proof that $\mathcal{H}$ as defined in Section 33.1 is complete.

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Problem 33.2

Show that if in Example 33.8 we let

T=\left\{ f \in C^{1}([0,1]) ;\| f\| _{\infty } \leq 1,\left\| f^{\prime }\right\| _{\infty } \leq 1\right\}

then $N(\varepsilon )$ is bounded above by $c_{1} \varepsilon^{-1}$ and bounded below by $c_{2} \varepsilon^{-1}$ .

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Problem 33.3

Suppose $X^{i}$ and $Y^{i}$ are two sequences of Brownian motions with all of the Brownian motions independent of each other. Let

Z_{(s, t)}^{n}=\frac{1}{\sqrt{n}} \sum _{i=1}^{n} X_{s}^{i} Y_{t}^{i}

Prove that $Z^{n}$ converges weakly with respect to the topology of $C\left([0,1]^{2}\right)$ as $n \rightarrow \infty$ to a Brownian sheet.

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Problem 33.4

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

\operatorname {Cov}\left[ X_{s}, X_{t} \right]=s \wedge t-s t, \quad 0 \leq s, t \leq 1

Identify the reproducing kernel Hilbert space for $X$ .

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Problem 33.5

Let $X$ be the Ornstein-Uhlenbeck process started at 0 . This was defined in Exercise 19.5. Identify the reproducing kernel Hilbert space for $X$ .

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