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| For our first ] example, take the domain to be N, the ], and our range to be R, the ]. Then, a function f : N -> R is a ] of real ], and the following questions arise: | For our first ] example, take the domain to be N, the ], and our range to be R, the ]. Then, a function f : N -> R is a ] of real ], and the following questions arise: | ||
| # How is a ] specified? | # How is a ] specified? | ||
| # How do we find the answers to typical questions about sequences, such as | # How do we find the answers to typical questions about sequences, such as | ||
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| Another important class of examples is when the domain is not a ] such as the natural numbers, but a ] such as the unit interval , the positive real numbers ], R. In this case, we have a different set of questions that we might want to answer: | Another important class of examples is when the domain is not a ] such as the natural numbers, but a ] such as the unit interval , the positive real numbers ], R. In this case, we have a different set of questions that we might want to answer: | ||
| # How is a random function specified? | # How is a random function specified? | ||
| # How do we find the answers to typical questions about functions, such as | # How do we find the answers to typical questions about functions, such as | ||
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| In the ordinary ] of ] by means of ], the problem is to construct a ] of ] subsets of the space of all functions, and then put a finite ] on it. For this purpose one traditionally uses a method called ] extension. | In the ordinary ] of ] by means of ], the problem is to construct a ] of ] subsets of the space of all functions, and then put a finite ] on it. For this purpose one traditionally uses a method called ] extension. | ||
| The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of functions f : X -> Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables (f(x_1),...,f(x_n)). Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for (f(x_1),...,f(x_{n-1})). There is an obvious compatibility condition, namely, that the marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation. | The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of functions f : X -> Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables (f(x_1),...,f(x_n)). Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for (f(x_1),...,f(x_{n-1})). There is an obvious compatibility condition, namely, that the marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the ]. | ||
| Given a family of compatible finite-dimensional probability distributions, the Kolmogorov extension theorem guarantees the existence of a stochastic process with the given finite-dimensional probability distributions. | Given a family of compatible finite-dimensional probability distributions, the Kolmogorov extension theorem guarantees the existence of a stochastic process with the given finite-dimensional probability distributions. | ||
| ]] | |||
| ==== Separability, or what the Kolmogorov extension does not provide ==== | ==== Separability, or what the Kolmogorov extension does not provide ==== | ||
| Recall that, in the Kolmogorov axiomatization, ] are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a (probabilistic) answer. | |||
| The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates (f(x_1),...,f(x_n)) are restricted to lie in measurable subsets of Y^n. In other words, if a (yes/no) question can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer. | |||
| In measure theory, if we have a ] collection of measurable sets, then the union and intersection of al of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer. | |||
| This means that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, it means that every question that one could ask about a sequence has a probabilistic answer when applied to a random sequence. It also means that certain questions about functions on a continuous domain don't have a probabilistic answer. | |||
| One might hope that the questions that depend on uncountably many values of a function be of little interest, but virtually all concepts of calculus are of this sort. For example: | |||
| #the ] of a function on an interval | |||
| #limits of functions | |||
| #continuity | |||
| #differentiability | |||
| all require knowledge of uncountably many values of the function. | |||
Revision as of 08:59, 7 April 2002
A stochastic process is a random function. This means that, if
f : D -> R
is a random function with domain D and range R, the image of each point of D, f(x), is a random variable with values in R.
Of course, the mathematical definition of a function includes the case "a function from {1,...,n} to R is a vector in R^n", so multidimensional random variables are a special case of stochastic processes.
For our first infinite example, take the domain to be N, the natural numbers, and our range to be R, the real numbers. Then, a function f : N -> R is a sequence of real numbers, and the following questions arise:
- How is a random sequence specified?
- How do we find the answers to typical questions about sequences, such as
- what is the probability distribution of the value of f(i)?
- what is the probability that f is bounded?
- what is the probability that is f monotonic?
- what is the probability that f(i) has a limit as i->infty?
- if we construct a series from f(i), what is the probability that the series converges? What is the probability distribution of the sum?
Another important class of examples is when the domain is not a discrete space such as the natural numbers, but a continuous space such as the unit interval , the positive real numbers [0,infty) or the entire real line, R. In this case, we have a different set of questions that we might want to answer:
- How is a random function specified?
- How do we find the answers to typical questions about functions, such as
- what is the probability distribution of the value of f(x)?
- what is the probability that f is bounded/integrable/continuous/differentiable...?
- what is the probability that f(x) has a limit as x->infty?
Constructing stochastic processes: the Kolmogorov extension
In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.
The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of functions f : X -> Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables (f(x_1),...,f(x_n)). Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for (f(x_1),...,f(x_{n-1})). There is an obvious compatibility condition, namely, that the marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.
Given a family of compatible finite-dimensional probability distributions, the Kolmogorov extension theorem guarantees the existence of a stochastic process with the given finite-dimensional probability distributions.
Separability, or what the Kolmogorov extension does not provide
Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a (probabilistic) answer.
The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates (f(x_1),...,f(x_n)) are restricted to lie in measurable subsets of Y^n. In other words, if a (yes/no) question can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.
In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of al of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.
This means that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, it means that every question that one could ask about a sequence has a probabilistic answer when applied to a random sequence. It also means that certain questions about functions on a continuous domain don't have a probabilistic answer.
One might hope that the questions that depend on uncountably many values of a function be of little interest, but virtually all concepts of calculus are of this sort. For example:
- the supremum of a function on an interval
- limits of functions
- continuity
- differentiability
all require knowledge of uncountably many values of the function.