Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 280

with the response of the structure and which are necessary to predict the collapse
probability?
In order to answer that question, we will first recap a few ideas about random
processes.
8.4. A few reminders about random processes
For more information, see [GIB 88], [PRE 90] and [KRE 83], though these are
more theoretical.
8.4.1. Definition and characterization of a time random process
Consider a set of time instants arranged in ascending order. Each is associated
with an X n random variable. The set of these variables constitutes a random process.
The process will thus be completely characterized by the joint repartition
functions and probability densities of the X n set. The discrete process defined above
can be more generally applied to a continuous process, denoted X(t), by having the
between t n intervals tend to zero. In that case, complete characterization obviously
requires an infinite amount of information.
From a practical point of view, we will only be able to consider a limited number
of joint probabilities, and therefore the characterization will be imperfect.
In most cases, considering only the p 2 (x 1 ,t 1 ; x 2 ,t 2 ) joint probability densities or
two X n variables will be enough. If the probability density completely characterizes
the process, the latter is said to be Markovian .
8.4.2. Second order characterization
Rather than using the p 2 function, we will consider the associated moment
function, limiting ourselves to order 2 (which, in the case of a Gaussian process, is
enough to characterize it). Hence if (D) is the set of values X can assume, we can
define:
(i) the process average;