CHAPTER 12 
PROBABILISTIC CHARACTERS OF NONLINEAR RESPONSE PROCESS 
12.1 INTRODUCTION 
The stochastic or random process can be expressed as a function of tme and also as 
a probabilisitc measure x(t,s). Until Chapter 11, however, we have discussed the charac- 
ters of the process mostly in the time aspect and, based on the ergodicity, dealt with the 
correlation functions and spectra and their statistical characteristics. Especially in 
Chapters 9, 10, and 11, the effects of nonlinearities on these characteristics were studied. 
In this chapter, the probability distributions that characterize the response process 
will be discussed briefly for reference. The main topic in Part II is the nonlinearity of the 
process. However, general considerations, including the linear cases, will be discussed 
first. 
Generally, the probabilistic character of the random process X(t) can be defined 
completely by the series of probability distribution density functions as 
Pi(x1,t;)dx : the probability that X(t) is in the range x) — x; + dx 
at tme 14; 
P2(%1,t1; X2,t2) : the probability that X(t) is in the range x) — x1 + ax 
at time 7 
and in x2 — X2+ dx at time 19; 
P3(X1, t1; X2, 12; X3, 3) : the probability that X(t) is in the range xj — x) + dx) 
at time 7), 
in X2 — X2+dx> at time fp, 
and in x3 — x3+dx3 at time f3. 
In the same way p, is defined for n=4,5... . 
Here i. p, 20 
ii. Pp 1S Symmetrical for x), tf); X2, t23X3, 13; ... 
iii. the marginal probability density function p,, is 
