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A stochastic process is a . This means that, if A ] is a ]. This means that, if
f : D -> R f : D -> R
is a random function with D and R, the image of each point of D, f(x), is a with values in R. is a random function with ] D and ] R, the image of each point of D, f(x), is a ] with values in R.
Of course, the mathematical definition of a includes the case "a function from {1,...,n} to R is a in R^n", so multidimensional random variables are a special case of stochastic processes. Of course, the mathematical definition of a ] includes the case "a function from {1,...,n} to R is a ] in R^n", so multidimensional random variables are a special case of stochastic processes.
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:
1) How is a specified? # How is a ] specified?
⚫ # How do we find the answers to typical questions about sequences, such as
⚫ ## what is the of the value of f(i)?
⚫ 2) How do we find the answers to typical questions about sequences, such as
⚫ ## what is the that f is ?
a) what is the of the value of f(i)? ## what is the probability that is f ?
## what is the probability that f(i) has a as i->infty?
b) what is the that f is ? ## if we construct a from f(i), what is the probability that the series ? What is the probability of the sum?
⚫ c) what is the probability that is f ?
⚫ d) what is the probability that f(i) has a as i->infty?
e) if we construct a from f(i), what is the probability that the series ? What is the probability of the sum?

Revision as of 07:41, 6 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:

  1. How is a random sequence specified?
  2. How do we find the answers to typical questions about sequences, such as
    1. what is the of the value of f(i)?
    2. what is the that f is ?
    3. what is the probability that is f ?
    4. what is the probability that f(i) has a as i->infty?
    5. if we construct a from f(i), what is the probability that the series ? What is the probability of the sum?