User:DarenCline/sigma algebra
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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
[edit]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
[edit]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 has finite expectation. A random variable is the conditional expectation of with respect to a σ-algebra , and typically denoted by , if
- is measurable with respect to : , and
- for all ,
where is the indicator function of the set . 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:
When is the σ-algebra generated by a random variable (or vector, or process) , it is usual to express the conditional expectation as .
Conditional expectation has many useful properties; a few of the more basic ones showing the roles of σ-algebras are recounted here.
- If is independent of all then with probability 1.
- If is measurable with respect to then with probability 1.
- (Tower) If is a σ-algebra such that then with probability 1.
Martingales and Markov Processes
[edit]The following is a short description of the uses of ordered collections of σ-algebras for certain types of stochastic processes.
Suppose (usually {0, 1, 2, …} or (0, ∞)), is a probability space and is a stochastic process.
- A filtration is a collection of σ-algebras such that each and s < t implies .
- The natural filtration for is given by , that is, the σ-algebra generated by the process up to and including time t.
- is adapted to a filtration if its natural filtration satisfies for all .
Filtrations are important for conditioning on the past behavior of a process.
is called a martingale with respect to if is adapted to and s < t implies
If 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.
is called a Markov process if s < t implies
Moreover, is said to be homogeneous if this is a function only of , t − s, and .
The Markov property just described has equivalent generalizations. For example, it implies
whenever h is a bounded function from to and .
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
[edit]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, is a probability space, which means is a σ-algebra of subsets of and 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 and limn→∞ An exists then
Borel-Cantelli Lemmas
[edit]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
The second Borel-Cantelli lemma is a partial converse:
Almost Sure Convergence
[edit]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 . It would be a mistake, however, to write this simply as a limsup of events. Instead, the complement of the event is
Therefore,
can be replaced with a more general measure μ, in which case is a measure space.