Stochastic process, generalized

A stochastic process $X$ depending on a continuous (time) argument $t$ and such that its values at fixed moments of time do not, in general, exist, but the process has only "smoothed values" $X(\varphi )$ describing the results of measuring its values by means of all possible linear measuring devices with sufficiently smooth weight function (or impulse transition function) $\varphi (t)$. A generalized stochastic process $x(\varphi )$ is a continuous linear mapping of the space $D$ of infinitely-differentiable functions $\varphi$ of compact support (or any other space of test functions used in the theory of generalized functions) into the space $L_0$ of random variables $X$ defined on some probability space. Its realizations $x(\varphi )$ are ordinary generalized functions of the argument $t$. Ordinary stochastic processes $X(t)$ can also be regarded as generalized stochastic processes, for which

$$X(\varphi )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\varphi (t)X(t)dt;$$

this is particularly useful in combination with the fact that a generalized stochastic process $X$ always has derivatives $X^{(n)}$ of any order $n$, given by

$$X^{(n)}(\varphi )=(-1{)}^{n}X({\varphi }^{(n)})$$

(see, for example, Stochastic process with stationary increments). The most important example of a generalized stochastic process of non-classical type is that of white noise. A generalization of the concept of a generalized stochastic process is that of a generalized random field.

For references, see Random field, generalized.

Comments[edit]

References[edit]