Pm = | see | p. dXm+1dXm+2 016 6 Gx (12.1) 
As was mentioned in Section 2.1, when these probability density functions p, stay 
the same when the time 1; is replaced by 1,+T7 (T: arbitrary), this process is called station- 
ary. Then pj(xj, t}) is independent of time, and p> depends only on the time difference 
t7-t). Here the concept of the Markov process that is used to classify the random process 
probability is first introduced. 
From the characteristics of these distribution functions, the processes are sometimes 
classified into three groups: (1) pure random processes, (2) Markov processes, and (3) 
general processes. 
For a pure random process, since the value x; at time 7 is independent of any values 
of x(t) at any other times 4-1, f;-2, . . . t2, 41, this process can be defined completely by 
P1(%,t;), because any higher order joint probability distribution function is 
n 
Pn(X1, 13 X2,t2 - - - Xn tn) = _ PGi, ti) . (12.2) 
I= 
Accordingly, if only p;(%;,t;) is given, all other joint distribution functions can be derived 
easily. 
12.2 MARKOV PROCESS 
When the process is not purely random, but when in addition to p like p,(x;,t;) and 
PiQi-1,ti-1), if p2(%-1, tj-1; xj, tj) 1s necessary to express the probability characters of the 
process, the process is called a Markov (or Markovian) (linear) process. The condition 
that the probability distribution density functions p(x;-1, t-1) and p2(%}-1, t}-1; %j, tj) are 
given is the same as the condition that the conditional probability distribution density 
functions p-,(x;, tj — t;-11xj-1) are known. Because the conditional probability density func- 
tion Po,(%;, tlxj-t}-1) is the probability distribution density function of this process that x 
is in the range x; — x;+ dx; at time ¢;, under the condition that the value of this process 
was in the range of xj) — xj-1 + dx; at time 4,4, thatis (¢;— 7, ) prior to 7, and is 
expressed by 
P2(X;, tj, Xj-1,t;-1) 
(12.3) 
PiQr 1) 
Dey (Xj, til Xj-1, ti-1) = Po,(Xjp tj — Hailxp) = 
Here, from the characteristics of the probability distribution density function, 
1. PcAXjp tlx;-1) =>0 
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