User:DarenCline/sigma algebra

The following is a draft of changes/additions I am considering for the articles on σ-algebras and set-theoretic limit. I especially have in mind including examples of their use in probability.

Special Uses in Probability

This section demonstrates some of the important uses of σ-algebras in probability beyond what has been described above. It does not, however, do this thoroughly; see the relevant articles instead.

Conditional Expectation

Conditional expectation refers to a prediction of one random variable on the basis of given values of one or more other random variables. It can be, and is, defined in a variety of ways including as the expectation of a conditional distribution and as a projection in the Hilbert space of random variables with finite second moment. The broadest definition, and the one most useful for proofs, uses a sub σ-algebra to represent the partial information that one is conditioning on. The discussion here is limited to demonstrating this role of σ-algebras.

The definition is a follows. Suppose $Y$ has finite expectation. A random variable $W$ is the conditional expectation of $Y$ with respect to a σ-algebra ${\mathcal {G}}$, and typically denoted by $\mathbb {E} (Y\mid {\mathcal {G}})$, if

1. $W$ is measurable with respect to ${\mathcal {G}}$: $\sigma (W)\subset {\mathcal {G}}$, and
2. $\mathbb {E} (W1_{A})=\mathbb {E} (Y1_{A})$ for all $A\in {\mathcal {G}}$,

where $1_{A}$ is the indicator function of the set $A$. This definition is not entirely unique: any two "versions" will be equal with probability 1. (This definition also does not describe how to "compute" the conditional expectation; that is left to other definitions and to use of properties of conditional expectations.)

Conditional probability is defined as a conditional expectation:

$\mathbb {P} (Y\in B\mid {\mathcal {G}})=\mathbb {E} (1_{Y\in B}\mid {\mathcal {G}}).$

When ${\mathcal {G}}$ is the σ-algebra generated by a random variable (or vector, or process) $X$, it is usual to express the conditional expectation as $\mathbb {E} (Y\mid X)$.

Conditional expectation has many useful properties; a few of the more basic ones showing the roles of σ-algebras are recounted here.

• If $Z$ is independent of all $A\in {\mathcal {G}}$ then $\mathbb {E} (ZY\mid {\mathcal {G}})=\mathbb {E} (Z)\,\mathbb {E} (Y\mid {\mathcal {G}})$ with probability 1.
• If $Z$ is measurable with respect to ${\mathcal {G}}$ then $\mathbb {E} (ZY\mid {\mathcal {G}})=Z\,\mathbb {E} (Y\mid {\mathcal {G}})$ with probability 1.
• (Tower) If ${\mathcal {F}}$ is a σ-algebra such that ${\mathcal {F}}\subset {\mathcal {G}}$ then $\mathbb {E} (\mathbb {E} (Y\mid {\mathcal {G}})\mid {\mathcal {F}})=\mathbb {E} (Y\mid {\mathcal {F}})$ with probability 1.

Martingales and Markov Processes

The following is a short description of the uses of ordered collections of σ-algebras for certain types of stochastic processes.

Suppose $\mathbb {T} \subset \mathbb {R}$ (usually {0, 1, 2, …} or (0, ∞)), $(\Omega ,\Sigma ,\mathbb {P} )$ is a probability space and $Y=\{Y_{t}\}_{t\in \mathbb {T} }$ is a stochastic process.

• A filtration is a collection of σ-algebras $\{{\mathcal {G}}_{t}\}_{t\in \mathbb {T} }$ such that each ${\mathcal {G}}_{t}\subset \Sigma$ and s < t implies ${\mathcal {G}}_{s}\subset {\mathcal {G}}_{t}$.
• The natural filtration for $Y$ is given by ${\mathcal {F}}_{t}=\sigma (\{Y_{t}\}_{s\leq t})$, that is, the σ-algebra generated by the process up to and including time t.
• $Y$ is adapted to a filtration $\{{\mathcal {G}}_{t}\}$ if its natural filtration satisfies ${\mathcal {F}}_{t}\subset {\mathcal {G}}_{t}$ for all $t\in \mathbb {T}$.

Filtrations are important for conditioning on the past behavior of a process.

$Y$ is called a martingale with respect to $\{{\mathcal {G}}_{t}\}$ if $Y$ is adapted to $\{{\mathcal {G}}_{t}\}$ and s < t implies

$\mathbb {E} (Y_{t}\mid {\mathcal {G}}_{s})=Y_{s}.$

If $Y$ is a martingale with respect to any filtration then it also is a martingale with respect to its natural filtration, a result which can be demonstrated with the tower property.

$Y$ is called a Markov process if s < t implies

$\mathbb {P} (Y_{t}\in A\mid {\mathcal {F}}_{s})=\mathbb {P} (Y_{t}\in A\mid \sigma (Y_{s})).$

Moreover, $Y$ is said to be homogeneous if this is a function only of $A$, t − s, and $Y_{s}$.

The Markov property just described has equivalent generalizations. For example, it implies

$\mathbb {E} (h(Y_{t_{1}},\dots ,Y_{t_{n}})\mid {\mathcal {F}}_{t})=\mathbb {E} (h(Y_{t_{1}},\dots ,Y_{t_{n}})\mid \sigma (Y_{t}))$

whenever h is a bounded function from $\mathbb {R} ^{n}$ to $\mathbb {R}$ and $t<t_{1}<\cdots <t_{n}$.

A martingale need not be a Markov process, nor does a Markov process have to be a martingale. However, many important results can be proved by deriving a martingale from a Markov process.

Probability Uses for Limits of Sets

Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. Such limits are used to calculate (or prove) the probabilities and measures of other, more purposeful, sets. For the following, $\scriptstyle (X,{\mathcal {F}},\mathbb {P} )$ is a probability space, which means $\scriptstyle {\mathcal {F}}$ is a σ-algebra of subsets of $\scriptstyle X$ and $\scriptstyle \mathbb {P}$ is a probability measure defined on that σ-algebra. Sets in the σ-algebra are known as events.

If A₁, A₂, ... is a sequence of events in $\scriptstyle {\mathcal {F}}$ and lim_n→∞ A_n exists then

$\mathbb {P} (\lim _{n\rightarrow \infty }A_{n})=\lim _{n\rightarrow \infty }\mathbb {P} (A_{n}).$

Borel-Cantelli Lemmas

In probability, the two Borel-Cantelli Lemmas can be useful for showing that the limsup of a sequence of events has probability equal to 1 or to 0. The statement of the first (original) Borel-Cantelli lemma is

$\sum _{n=1}^{\infty }\mathbb {P} (A_{n})<\infty \Rightarrow \mathbb {P} (\limsup _{n\rightarrow \infty }A_{n})=0.$

The second Borel-Cantelli lemma is a partial converse:

$A_{1},A_{2},\dots \ {\mbox{are independent events and}}\ \sum _{n=1}^{\infty }\mathbb {P} (A_{n})=\infty \Rightarrow \mathbb {P} (\limsup _{n\rightarrow \infty }A_{n})=1.$

Almost Sure Convergence

One of the most important applications to probability is for demonstrating the almost sure convergence of a sequence of random variables. The event that a sequence of random variables Y₁, Y₂, ... converges to another random variable Y is formally expressed as $\scriptstyle \{\limsup _{n\to \infty }|Y_{n}-Y|=0\}$. It would be a mistake, however, to write this simply as a limsup of events. Instead, the complement of the event is

$\{\limsup _{n\to \infty }|Y_{n}-Y|\neq 0\}=\{\limsup _{n\to \infty }|Y_{n}-Y|>{\frac {1}{k}}\ {\mbox{for some}}\ k\}$

$=\bigcup _{k\geq 1}\bigcap _{n\geq 1}\bigcup _{j\geq n}\{|Y_{n}-Y|>{\frac {1}{k}}\}=\lim _{k\to \infty }\limsup _{n\to \infty }\{|Y_{n}-Y|>{\frac {1}{k}}\}.$

Therefore,

$\mathbb {P} (\{\limsup _{n\to \infty }|Y_{n}-Y|\neq 0\})=\lim _{k\to \infty }\mathbb {P} (\limsup _{n\to \infty }\{|Y_{n}-Y|>{\frac {1}{k}}\}).$

$\scriptstyle \mathbb {P}$ can be replaced with a more general measure μ, in which case $\scriptstyle (X,{\mathcal {F}},\mu )$ is a measure space.