Math Problems Vault

Problem 1.1

(a)(b)(c)

Let $X, Y, X_{n}, Y_{n}: \Omega \rightarrow \mathbb {R}, n \geqslant 1$ , be random variables.

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(a)

If, for all $n \geqslant 1, X_{n} \; \perp \! \! \! \perp \; Y_{n}$ and if $\left(X_{n}, Y_{n}\right) \xrightarrow {d}(X, Y)$ , then $X \; \perp \! \! \! \perp \; Y$ .

(b)

Let $X \; \perp \! \! \! \perp \; Y$ such that $X, Y \sim \beta_{1 / 2}: = \frac{1}{2}\left(\delta_{0}+\delta_{1}\right)$ are Bernoulli random variables. We set $X_{n}: = X+\frac{1}{n}$ and $Y_{n}: = 1 - X_{n}$ . Then $X_{n} \xrightarrow {d} X, Y_{n} \xrightarrow {d} Y, X_{n}+Y_{n} \xrightarrow {d} 1$ but $\left(X_{n}, Y_{n}\right)$ does not converge weakly to $(X, Y)$ .

(c)

Assume that $X_{n} \xrightarrow {d} X$ and $Y_{n} \xrightarrow {d} Y$ . Is it true that $X_{n}+Y_{n} \xrightarrow {d} X+Y$ ?

Problem 1.2

Let $X_{n}, Y_{n}: \Omega \rightarrow \mathbb {R}^{d}, n \geqslant 1$ , be two sequences of random variables such that $X_{n} \xrightarrow {d} X$ and $X_{n} - Y_{n} \xrightarrow {\mathbb {P}} 0$ . Then $Y_{n} \xrightarrow {d} X$ .

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Problem 1.3

(a)(b)

Let $X_{n}, Y_{n}: \Omega \rightarrow \mathbb {R}, n \geqslant 1$ , be two sequences of random variables.

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(a)

If $X_{n} \xrightarrow {d} X$ and $Y_{n} \xrightarrow {\mathbb {P}} c$ , then $X_{n} Y_{n} \xrightarrow {d} c X$ . Is this still true if $Y_{n} \xrightarrow {d} c$ ?

(b)

If $X_{n} \xrightarrow {d} X$ and $Y_{n} \xrightarrow {\mathbb {P}} 0$ , then $X_{n}+Y_{n} \xrightarrow {d} X$ . Is this still true if $Y_{n} \xrightarrow {d} 0$ ?

Remark.

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The direct approach to (a) is possible but relatively ugly. Part (b) has a relatively simple direct proof: fix $\xi \in \mathbb {R}$ ,

\mathbb {E}\left[e^{i \xi (X_{n}+Y_{n})}\right] - \mathbb {E}\left[e^{i \xi X}\right] = \left(\mathbb {E}\left[e^{i \xi (X_{n}+Y_{n})}\right] - \mathbb {E}\left[e^{i \xi X_{n}}\right]\right)+\underbrace{\left(\mathbb {E}\left[e^{i \xi X_{n}}\right] - \mathbb {E}\left[e^{i \xi X}\right]\right)}_{\to 0 \text{ by } d\text{-convergence}}.

For the first term, using the uniform-continuity argument from Problem 1.2, for any $\varepsilon > 0$ there is $\delta = \delta (\varepsilon , \xi ) > 0$ with

\left|\mathbb {E}\left[e^{i \xi (X_{n}+Y_{n})}\right] - \mathbb {E}\left[e^{i \xi X_{n}}\right]\right| \leqslant \mathbb {E}\left[\left|e^{i \xi X_{n}}\left(e^{i \xi Y_{n}} - 1\right)\right|\right] = \mathbb {E}\left[\left|e^{i \xi Y_{n}} - 1\right|\right] \leqslant \varepsilon +\mathbb {P}\left(\left|Y_{n}\right| > \delta \right) \xrightarrow [n \rightarrow \infty ]{} \varepsilon \xrightarrow [\varepsilon \rightarrow 0]{} 0,

using $Y_{n} \xrightarrow {\mathbb {P}} 0$ in the penultimate step.

Problem 1.4

Let $X_{n}, X, Y: \Omega \rightarrow \mathbb {R}, n \geqslant 1$ , be random variables. If

\lim _{n \rightarrow \infty } \mathbb {E}\left[f\left(X_{n}\right) g(Y)\right] = \mathbb {E}\left[f(X) g(Y)\right] \quad \text{ for all } f \in \mathcal{C}_{b}(\mathbb {R}), g \in \mathcal{B}_{b}(\mathbb {R}) \text{, }

then $\left(X_{n}, Y\right) \xrightarrow {d}(X, Y)$ . If $X = \phi (Y)$ for some $\phi \in \mathcal{B}(\mathbb {R})$ , then $X_{n} \xrightarrow {\mathbb {P}} X$ .

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Problem 1.5

(a)(b)(c)

Let $\delta_{j}, j \geqslant 1$ , be iid Bernoulli random variables with $\mathbb {P}\left(\delta_{j} = \pm 1\right) = 1 / 2$ . We set

S_{0}: = 0, \quad S_{n}: = \delta _{1}+\cdots +\delta _{n} \quad \text{ and } \quad X_{t}^{n}: = \frac{1}{\sqrt{n}} S_{\left\lfloor n t\right\rfloor }

A one - dimensional random variable $G$ is Gaussian if it has the characteristic function $\mathbb {E}\left[e^{i \xi G}\right] = \exp \left(i m \xi - \frac{1}{2} \sigma^{2} \xi^{2}\right)$ with $m \in \mathbb {R}$ and $\sigma \geqslant 0$ . Prove that

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(a)

$X_{t}^{n} \xrightarrow {d} G_{t}$ where $t > 0$ and $G_{t}$ is a Gaussian random variable.

(b)

$X_{t}^{n} - X_{s}^{n} \xrightarrow {d} G_{t - s}$ where $t \geqslant s \geqslant 0$ and $G_{u}$ is a Gaussian random variable. Do we have $G_{t - s} = G_{t} - G_{s}$ ?

(c)

Let $0 \leqslant t_{1} \leqslant \ldots \leqslant t_{m}, m \geqslant 1$ . Determine the limit as $n \rightarrow \infty$ of the random vector $\left(X_{t_{m}}^{n} - X_{t_{m - 1}}^{n}, \ldots , X_{t_{2}}^{n} - X_{t_{1}}^{n}, X_{t_{1}}^{n}\right)$ .

Problem 1.6

Consider the condition for all $s < t$ the random variables $\frac{B(t) - B(s)}{\sqrt{t - s}}$ are identically distributed, centred and have variance 1.

Show that (B0), (B1), (B2), (B3) and (B0), (B1), (B2), (B3') are equivalent.

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Problem 1.7

Let $G$ and $G^{\prime }$ be two independent real N $(0,1)$ - random variables. Show that

\Gamma ^{ \pm }: = \frac{G \pm G^{\prime }}{\sqrt{2}} \sim \mathrm{N}(0,1) \quad \text{ and } \quad \Gamma ^{ - } \; \perp \! \! \! \perp \; \Gamma ^{+}

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Problem 1.8

(a)(b)

Let $\left(X_{t}\right)_{t \geqslant 0}$ be a stochastic process on the measure space $(\Omega , \mathscr {A}, \mathbb {P})$ satisfying (B1)(B3). Assume further that there is some measurable set $\Omega_{0} \subset \Omega , \mathbb {P}\left(\Omega_{0}\right) = 1$ , such that for all $\omega \in \Omega_{0}$ we have $X_{0}(\omega ) = 0$ and $t \mapsto X_{t}(\omega )$ is continuous. Show that

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(a)

$B_{t}(\omega ): = X_{t}(\omega ) \; \mathbb {1}_{\Omega_{0}}(\omega )$ is a Brownian motion on $(\Omega , \mathscr {A}, \mathbb {P})$ ;

(b)

$X_{t}$ is a Brownian motion on the space $\left(\Omega_{0}, \Omega_{0} \cap \mathscr {A}, \mathbb {P}_{0}: = \mathbb {P}\left(\cdot \cap \Omega_{0}\right)\right)$ .

Notation. $\Omega_{0} \cap \mathscr {A}: = \left\{ \Omega_{0} \cap A: A \in \mathscr {A}\right\}$ is the trace $\sigma$ - algebra.

Remark.

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Method (b) also works if $\Omega_{0}$ is not measurable, see Problem 1.9.

Problem 1.9

Let $(\Omega , \mathscr {A}, \mathbb {P})$ be a probability space and let $\mathbb {P}^{*}\left(Q\right): = \inf \left\{ \mathbb {P}\left(A\right): A \supset Q, A \in \mathscr {A}\right\}$ be the outer measure. If $\Omega_{0} \subset \Omega$ is such that $\mathbb {P}^{*}\left(\Omega_{0}\right) = 1$ , then $\left(\Omega_{0}, \mathscr {A}_{0}, \mathbb {P}_{0}\right)$ with $\mathscr {A}_{0}: = \Omega_{0} \cap \mathscr {A}$ and $\mathbb {P}_{0}\left(A_{0}\right): = \mathbb {P}\left(A\right)$ for every $A_{0} \in \mathscr {A}_{0}$ of the form $A_{0} = \Omega_{0} \cap A$ , defines a new probability space.

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Problem 1.10

Let $(\Omega , \mathscr {A}, \mathbb {P})$ be a probability space. The completion $\left(\Omega , \mathscr {A}^{\prime }, \mathbb {P}^{\prime }\right)$ is the smallest extension such that $\mathscr {A}^{\prime }$ contains all subsets of $\mathscr {A}$ measurable null sets. Show that

\begin{aligned} \mathscr {A}^{\prime }: = & \left\{ A^{\prime } \subset \Omega : \exists A_{1}, A_{2} \in \mathscr {A}, A_{1} \subset A^{\prime } \subset A_{2}, \mathbb {P}\left(A_{2} \backslash A_{1}\right) = 0\right\} , \\ & \mathbb {P}^{\prime }\left(A^{\prime }\right): = \mathbb {P}\left(A_{2}\right) \quad \left(A_{2} \text{ is as in the definition of } \mathscr {A}^{\prime }\right) . \end{aligned}

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