Basic
Properties of Stochastic Processes

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**Definition of Stochastic Processes**

A stochastic process is a collection of random variables

{X(t), t T}

all defined on a common sample (probability) space. The X(t) is the state while (time) t is the index which is a member of set T.

*Examples* are the delay {D(i), i = 1, 2,
...} of the i-th customer and number of customers {Q(t), T 0}
in the queue at time t in a queue. In the first example, we have a discrete
time, continuous state, while in the second example the state is discrete and
time in continuous.

To perform statistical analysis of the simulation output we need to establish some conditions, e.g. output data must be a covariance stationary process (e.g. the data collected over n simulation runs).

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**Stationary Process **

A stationary stochastic process is a stochastic process {X(t), t T} with the property that the joint distribution all vectors of h dimension remain the same for any fixed h.

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A stochastic
process is *a first order stationary*
if expected of X(t) remains the same for all t.

For example in economic time series, a process is first order stationary when we remove any kinds of trend by some mechanisms such as differencing.

A stochastic
process is a *second order stationary*
if it is first order stationary and covariance between X(t) and X(s) is
function of t-s only.

Again, in economic time series, a process is second order stationary when we stabilise also its variance by some kind of transformations such as taking square root.

Clearly, a stationary process is a second order stationary, however the reverse may not hold.

In simulation
output statistical analysis we are satisfied if the output is *covariance stationary*.

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**Covariance Stationary**

A covariance
stationary process is a stochastic process {X(t), t T}
having finite second moments, i.e. expected of [X(t)]^{2} be finite.

Clearly, any stationary process with finite second moment is covariance stationary. A stationary process may have no finite moment whatsoever.

Since a Gaussian process needs a mean and covariance matrix only, it is stationary (strictly) if it is covariance stationary.