Product spaces and independence

Show that $\mathbb {P}\left(B_{1}^{c} B_{2}^{c} \ldots B_{n}^{c}\right) \leq \exp \left(-\sum_{i = 1}^{n} \mathbb {P}\left(B_{i}\right)\right) \rightarrow 0$.

Deduce that $\prod_{i = 1}^{\infty } B_{i}^{c} = 0$ almost surely.

Deduce that $\sum_{i = 1}^{\infty } B_{i} \geq 1$ almost surely.

Deduce that $\sum_{i = m}^{\infty } B_{i} \geq 1$ almost surely, for each finite $m$. Hint: The events $B_{m}, B_{m+1}, \ldots$ are independent.

Complete the proof.

The map $\phi : \mathbb {R}^{2} \rightarrow \mathbb {R}$ defined by $\phi (s, t) = \left\{ s > t\right\}$ is $\mathcal{B}\left(\mathbb {R}^{2}\right) \backslash \mathcal{B}(\mathbb {R})$-measurable. (What do you know about inverse images for $\phi$ ?)

Prove that the map $(x, y) \mapsto (f(x), y)$ is $\mathcal{A} \otimes \mathcal{B}(\mathbb {R}) \backslash \mathcal{B}(\mathbb {R}) \otimes \mathcal{B}(\mathbb {R})$-measurable.

Show that the composition of measurable functions is measurable, if the various sigma-fields fit together in the right way.

Complete the argument.

Show that $\mathcal{B}(X) \otimes \mathcal{B}(y) \subseteq \mathcal{B}(X \times y)$.

If both $X$ and $y$ have countably generated topologies, prove equality of the two sigma-fields on the product space.

Show that $\mathcal{B}\left(\mathbb {R}^{n}\right) = \mathcal{B}\left(\mathbb {R}^{k}\right) \otimes \mathcal{B}\left(\mathbb {R}^{n-k}\right)$.

Show that the Borel sigma-field $\mathcal{B}(X \times X)$ consists of all subsets of $X \times X$.

Show that $\mathcal{B}(X) \otimes \mathcal{B}(X)$ equals

For a given countable class of subsets $\mathcal{C}$ of $X$ define an equivalence relation $x \sim y$ if and only if $\left\{ x \in C\right\} = \left\{ y \in C\right\}$ for all $C$ in $\mathcal{C}$. Show that there are at most $2^{\kappa_{0}}$ equivalence classes. Deduce that there exists at least one pair of distinct points $x_{0}$ and $y_{0}$ such that $x_{0} \sim y_{0}$. Deduce that $\sigma (\mathcal{C})$ cannot separate the pair ($x_{0}, y_{0}$).

Show that the diagonal $\Delta = \left\{ (x, y) \in X \times X: x = y\right\}$ cannot belong to the product sigma-field $\mathcal{B}(X) \otimes \mathcal{B}(X)$, which is therefore a proper sub-sigma-field of $\mathcal{B}(X \times X)$. Hint: Suppose $\Delta \in \sigma (\mathcal{E})$ for some countable class $\mathcal{E}$ of measurable rectangles. Find a pair of distinct points $x_{0}, y_{0}$ such that no member of $\mathcal{E}$, or of $\sigma (\mathcal{E})$, can extract a proper subset of $F = \left\{ \left(x_{0}, x_{0}\right),\left(y_{0}, x_{0}\right),\left(x_{0}, y_{0}\right),\left(y_{0}, y_{0}\right)\right\}$, but $F \Delta = \left\{ \left(x_{0}, x_{0}\right),\left(y_{0}, x_{0}\right)\right\}$.

For fixed $E_{2}, \ldots , E_{n}$ in $\mathcal{E}_{2}, \ldots , \mathcal{E}_{n}$, define $\mathcal{D}_{1}$ as the class of all events $D$ for which $\mathbb {P}\left(D E_{2} \ldots E_{n}\right) = \mathbb {P}\left(D\right)\mathbb {P}\left(E_{2}\right) \ldots \mathbb {P}\left(E_{n}\right)$. Show that $\mathcal{D}_{1}$ is a $\lambda$-class.

Deduce that $\mathcal{D}_{1} \supseteq \sigma \left(\mathcal{E}_{1}\right)$. That is, that the analog of the hypothesized factorization with $\mathcal{E}_{1}$ replaced by $\sigma \left(\mathcal{E}_{1}\right)$ also holds.

Argue similarly that each subsequent $\mathcal{E}_{i}$ can also be replaced by its $\sigma \left(\mathcal{E}_{i}\right)$.

Use the Kolmogorov zero-one law (Example $\langle 12 \rangle$) to show that the set $\left\{ \lim \sup \bar{Z}_{n} > r\right\}$ is a tail event for each constant $r$, and hence it has probability either zero or one. Deduce that $\limsup \bar{Z}_{n} = c_{0}$ almost surely, for some constant $c_{0}$ (possibly $\pm \infty$).

If $\bar{Z}_{n}$ converges to a finite limit (possibly random) at each $\omega$ in a set $A$ with $\mathbb {P}\left(A\right) > 0$, show that in fact there must exists a finite constant $c_{0}$ for which $\bar{Z}_{n} \rightarrow c_{0}$ almost surely.

Show that there are at most countably many points $x$, atoms, for which $\mu \left\{ x\right\} > 0$.

Show that

where $\left\{ x_{i}: i \in \mathbb {N}\right\}$ contains all the atoms for both measures.

Explain how (ii) is related to the integration-by-parts formula:

Hint: Read Section 3.4.

Express $b_{i}$ as $\alpha_{1}+\ldots +\alpha_{i}$, for a sequence $\left\{ \alpha_{i}\right\}$ of nonnegative numbers. By a change in the order of summation, show, for $m < n$, that $\sum_{i = 1}^{n} b_{i} x_{i}$ equals

Given $\epsilon > 0$, find an $m$ such that $\left|\sum_{i = p}^{n} x_{i}\right| < \epsilon$ whenever $m \leq p \leq n$.

With $m$ as in (ii), and $n \geq m$, show that $\left|\sum_{i = 1}^{n} b_{i} x_{i}\right| \leq \left|C_{m}\right|+\epsilon \sum_{j = 1}^{n} \alpha_{j}$.

Deduce the asserted convergence.

Show that $\sum_{i = 1}^{\infty } \mathbb {E}\left[Z_{i}^{2}\right] / i^{2 \beta } < \infty$ and $\sum_{i = 1}^{\infty } \mathbb {E}\left[\left|Z_{i}\right|\right] / i^{\beta } < \infty$.

Deduce that $\sum_{i = 1}^{\infty } Z_{i} / i^{\beta }$ is convergent almost surely.

Show that $\sum_{i = 1}^{\infty } \mathbb {P}\left(\left|Y_{i}\right|^{\alpha } > i\right) < \infty$.

Deduce that $\sum_{i = 1}^{\infty } Y_{i} / i^{\beta }$ is convergent almost surely.

Deduce via Kronecker's Lemma (Problem [22]) that $n^{-1 / \alpha }\left(Y_{1}+\ldots +Y_{n}\right)$ converges to 0 almost surely.

Argue that there is no loss of generality in assuming $X_{i} \geq 0$. Hint: Consider positive and negative parts.

Show that $\mathbb {E}\left[T_{n}\right] / n \rightarrow \mu$ as $n \rightarrow \infty$ and $\left(S_{n}-T_{n}\right) / n \rightarrow 0$ almost surely.

Show that $\operatorname {Var}\left[T_{n}\right] \leq \sum_{i} \mathbb {E}\left[X_{1}^{2} \left\{ X_{1} \leq i \leq n\right\} \right]$.

For a fixed $\rho > 1$, let $\left\{ k_{n}\right\}$ be an increasing sequence of positive integers such that $k_{n} / \rho^{n} \rightarrow 1$. Show that $\sum_{n}\left\{ i \leq k_{n}\right\} / k_{n}^{2} \leq C / i^{2}$ for each positive integer $i$, for some constant $C$.

Use parts (iii) and (iv), and Problem [25], to show that

Deduce via the Borel-Cantelli lemma that $\left(T_{k_{n}}-\mathbb {E}\left[T_{k_{n}}\right]\right) / k_{n} \rightarrow 0$ almost surely.

Deduce that $S_{k_{n}} / k_{n} \rightarrow \mu$ almost surely, as $n \rightarrow \infty$.

For each $\rho^{\prime } > \rho$, show that

when $n$ is large enough.

Deduce that $\lim \sup S_{m} / m$ and $\liminf S_{m} / m$ both lie between $\mu / \rho$ and $\mu \rho$, with probability one.

Cast out a sequence of negligible sets as $\rho$ decreases to 1 to deduce that $S_{m} / m \rightarrow \mu$ almost surely.

If $F_{1}$ and $F_{2}$ are sets in $\mathcal{F}$ for which $F_{1} X = F_{2} X$, show that $P F_{1} = P F_{2}$.

Write $\mathcal{A}$ for the collection of all sets of the form $F X$, with $F \in \mathcal{F}$. Show that $\mathcal{A}$ is a sigma-field (the so-called trace sigma-field).

Show that $\mathcal{M}^{+}(X, \mathcal{A}) = \left\{ \left.\bar{f}(x)\right|_{x}: \bar{f} \in \mathcal{M}^{+}(\Omega , \mathcal{F})\right\}$.

Show that $Q(F X): = P F$ is a well defined (by (i)) probability measure on $\mathcal{A}$.

Show that $Q f = P \bar{f}$, for each $\bar{f} \in \mathcal{M}^{+}(\Omega , \mathcal{F})$ whose restriction to $X$ equals $f$.

Show that the sets $A_{i}$, for $i \in \overline{\mathbb {N}}$, are disjoint, and $\cup_{i \geq 0} A_{i} = [0,1)$.

Let $D_{0}$ be a Borel measurable subset of $A_{0}$. Define $D_{i}$ as the set of points $x+r_{i}$, with $x \in D_{0}$ and addition carried out modulo 1. Show $m D_{0} = m D_{i}$ for each $i$, and $1 \geq \sum_{i \geq 0} \mathfrak {m} D_{i}$. Deduce that $m D_{0} = 0$.

Deduce that $A_{0}$ cannot be measurable (not even for the completion of the Borel sigma-field), for otherwise $[0,1)$ would be a countable union of Lebesgue negligible sets.

Suppose $D$ is a Borel measurable subset of $U_{i \leq n} A_{i}$ for some finite $n$. Show that $[0,1)$ contains countably many disjoint translations of $D$. Deduce that $\mathfrak {m} D = 0$.

Define $X_{n} = \cup_{i \geq n} A_{i}$. Deduce from (iv) that each $X_{n}$ has Lebesgue outer measure 1 (that is, $m B = 1$ for each Borel set $B \supseteq X_{n}$), but $\cap_{n} X_{n} = \emptyset$.

Show that each function $f$ in $\mathcal{M}^{+}\left(\Omega_{n}, \mathcal{F}_{n}\right)$ is the restriction of a function $\bar{f}$ in $\mathcal{M}^{+}\left([0,1)^{n}, \mathcal{B}^{n}\right)$ to $\Omega_{n}$.

Show that $\mathbb {E}_{n}\left[f\right]: = \mathfrak {m}^{t} \bar{f}(t, t, \ldots , t)$, for $f \in \mathcal{M}^{+}\left(\Omega_{n}, \mathcal{F}_{n}\right)$, defines a consistent family of finite dimensional distributions.

Let $g_{n}$ denote the indicator function of $\left\{ \omega_{n} \in \Omega_{n}: x_{1} = x_{2} = \ldots = x_{n}\right\}$, and let $f_{n}(\omega ) = g_{n}(\omega \mid n)$ be the corresponding functions in $\mathcal{H}^{+}$. Show that $\mathbb {E}_{n}\left[g_{n}\right] = 1$ for all $n$, even though $f_{n} \downarrow 0$ pointwise.

Deduce that the functional $\mathbb {P}$, as defined by $\langle 47 \rangle$, is not sigma-smooth at zero. (Definition 47: on the cone $\mathcal{H}^{+}: = \cup_{n} \mathcal{H}_{n}^{+}$ of functions on $\Omega$ that depend on only finitely many coordinates, define $\mathbb {P} h: = \mathbb {E}_{n}\left[g_{n}\right]$ whenever $h(\omega ) = g_{n}(\omega \mid n)$ for some $g_{n}$ in the cone $\mathcal{G}_{n}^{+}$ of bounded, nonnegative, $\mathcal{F}_{n}$-measurable functions on $\Omega_{n}$; consistency of the family $\left\{ \mathbb {P}_{n}\right\}$ makes this well defined. An increasing functional $T$ on $\mathcal{H}^{+}$ is $\sigma$-smooth at zero if $T h_{i} \downarrow 0$ for every sequence $\left\{ h_{i}\right\}$ in $\mathcal{H}^{+}$ with $1 \geq h_{i} \downarrow 0$ pointwise.)

Let $\left\{ z_{n}: n \in \mathbb {N}\right\}$ be a sequence with $z_{n} \in H_{n}$. Use compactness of each $K_{i}$ to find subsequences $\mathbb {N}_{1} \supseteq \mathbb {N}_{2} \supseteq \mathbb {N}_{3} \supseteq \ldots$ of $\mathbb {N}$ for which $y(i): = \lim_{n \in \mathbb {N}_{i}} \pi_{i} z_{n}$ exists, as a point of $K_{i}$. Define $\mathbb {N}_{\infty }$ to be the subsequence whose $i$ th member equals the $i$ th member of $\mathbb {N}_{i}$. Show that $\lim_{n \in \mathbb {N}_{\infty }} \pi_{i} z_{n} = y(i)$ for every $i$.

Show that the first $i$ components of $y(i+1)$ coincide with $y(i)$ for each $i$. Deduce that there exists a $y$ in $\Omega$ for which $\pi_{i} y = y(i) \in K_{i}$ for every $i$. Deduce that $y \in H$, and hence $H \neq \emptyset$.

Specialize to the case where $z_{n} \in H$ for every $n$. Show that there is a subsequence that converges to a point of $H$.

Show that $\mathcal{F} = \cup_{S} \mathcal{F}_{S}$, the union running over all countable subsets $S$ of $T$.

For each countable $S \subset T$, let $\mathbb {P}_{S}$ be a probability measure on $\mathcal{F}_{S}$, with the property that $\mathbb {P}_{S}$ equals the restriction of $\mathbb {P}_{S^{\prime }}$ to $\mathcal{F}_{S}$ if $S \subseteq S^{\prime }$. Show that $\mathbb {P}\left(F\right): = \mathbb {P}_{S}\left(F\right)$ for $F \in \mathcal{F}_{S}$ defines a countably additive probability measure on $\mathcal{F}_{T}$.