DES - Definition of 'Independent Stochastic Process'

Independent Stochastic Process:

A stochastic process is called independent if the random variables:

$$X(t_1),X(t_2),\dots ,X(t_h)$$

are independent, for any choice of :

$$t_1<t_2<\dots <t_h\in \mathcal{T},h=2,3,\dots$$

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~Example:

Assume that you observed the stochastic process up to time $t$, and you want to infer (probabilistically) the value of the process at time $t+\tau$, using additional information provided by the observation.

For an independent process where:

$$\begin{array}{rl}& P(X(t+\tau )\le \bar{x}\mid X(s)=x(s)\forall s\le t)=\\ & =P(X(t+\tau )\le \bar{x})\end{array}$$

The knowledge of the past and current values of the process does not change our inference with respect to the prior

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