# What is a random process

# What is a random process

# Abstract and Keywords

A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in application). The variable can have a discrete set of values at a given time, or a continuum of values may be available. Likewise, the time variable can be discrete or continuous. A stochastic process is regarded as completely described if the probability distribution is known for all possible sets of times. A stationary process is one which has no absolute time origin. All probabilities are independent of a shift in the origin of time. This chapter discusses multitime probability description, conditional probabilities, stationary, Gaussian, and Markovian processes, and the Chapman–Kolmogorov condition.

*Keywords:*
random process, stochastic process, time variable, probability distribution, multitime probability description, conditional probabilities, stationary process, Gaussian process, Markovian process, Chapman–Kolmogorov condition

# 2.1 Multitime probability description

A random or stochastic process is a random variable *X*(*t*), at each time *t*, that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable in application). The variable *X* can have a discrete set of values *x _{j}* at a given time

*t*, or a continuum of values

*x*may be available. Likewise, the time variable can be discrete or continuous.

A stochastic process is regarded as completely described if the probability distribution

*t*

_{1},

*t*

_{2}, …,

*t*] of times. Thus we assume that a set of functions

_{n}*X*must be related to each other as the evolution in time of a single “stochastic” process.

_{j}# 2.2 Conditional probabilities

The concept of conditional probability introduced in Section 1.8 immediately generalizes to the multivariable case. In particular, Eq. (1.82)

*x*to be an abbreviation for

_{j}*X*(

*t*). The variables are written in time sequence since we regard the probability of

_{j}*x*as conditional on the earlier time values

_{n}*x*

_{n−1},…,

*x*

_{1}.

# 2.3 Stationary, Gaussian and Markovian processes

A stationary process is one which has no absolute time origin. All probabilities are independent of a shift in the origin of time. Thus

*τ*= −

*t*

_{1}. Specifically, for a stationary process, we expect that

A Gaussian process is one for which the multivariate distributions *P _{n}*(

*x*

_{n}, x_{n−1}, …,

*x*

_{1}) are Gaussians for all

*n*. A Gaussian process may, or may not be stationary (and conversely).

A Markovian process is like a student who can remember only the last thing he has been told. Thus it is defined by

*x*is sensitive to the last known event

_{n}*x*

_{n−1}and forgets all prior events. For a Markovian process, the conditional probability formula, Eq. (2.5) specializes to

*p*(

*x*

_{1}) and the “transition probabilities”

*p*(

*x*|

_{j}*x*

_{j−1}). If the Markovian process is also (p.46) stationary, all

*p*(

*x*|

_{j}*x*

_{j−1}) are described by a single transition probability

*t*

_{j−1}.

# 2.4 The Chapman–Kolmogorov condition

We have just shown that a Markovian random process is completely characterized by its “transition probabilities” *p*(*x* _{2}|*x* _{1}). To what extent is *p*(*x* _{2}|*x* _{1}) arbitrary? This question may be answered by taking Eq. (2.4) for a general random process specializing to the three time case and dividing by *p*(*x* _{1}) to obtain

*x*

_{2}we obtain

*t*obey:

*w*is the transition probability per unit time and the second term has been added to conserve probability. It describes the particles that have not left the state (p.47)

_{a′a}*a*provided that

*t*=

*t*

_{0}+ Δ

*t*

_{0}, we can evaluate the right hand side of the Chapman–Kolmogorov condition to first order in Δ

*t*and Δ

*t*

_{0}:

*p*(

*a′ t*

_{0}+ Δ

*t*+ Δ

*t*

_{0}|

*a*

_{0},

*t*

_{0}) expected from Eq. (2.18).

Note, however, that this proof did not make use of the conservation condition, Eq. (2.19). This will permit us, in Chapter 8, to apply the Chapman–Kolmogorov condition to processes that are Markovian but whose probability is not normalized.