Trajectory of stationary stochastic process not in $\mathrm{L}^2$

Question

Let $(X(t))_{t\in T}$ with $T\subseteq \mathbb{R}^n$ be a (strictly) stationary stochastic process. In the book 'Fourier Analysis and Stochastic Processes' Pierre Brémaud states on p.157 (Chapter 3.4.1) that the trajectory of $(X(t))_{t\in T}$ is almost surely not in $\mathrm{L}^2(\mathbb R)$ (he assumes $T\subseteq \mathbb R$). Unfortunately, he does not include a proof. I've got two questions regarding his statement:

1. Does someone know where I can find a proof of this statement or how to proof it?
2. Does this statement also hold for wide-sense stationary processes, i.e. are the trajectories of wide-sense stationary processes almost surely not in $\mathrm{L}^2(\mathbb R)$?

I would be happy if someone could help! Thanks a lot!

Answer 1

Consider the case $T=\Bbb R$. Other choices of $T$ can be handled in like fashion, the key being that the Lebesgue measure of $\Bbb R$ is infinite. Let $v:=\Bbb E[X(t)^2]$, assumed finite and (to avoid triviality) strictly positive. Because $(X(t))$ is stationary, $v$ doesn't depend on $t$. Consider now the $L^2$ norm of the sample path $t\mapsto X(t)$. Define $$ Z(t):=\int_t^\infty X(s)^2\,ds. $$ By stationarity, $Z(t)$ has the same distribution as $Z(0)$ for each $t>0$. But you also have $$ \lim_{t\to\infty}Z(t) = 0, $$ at each point of the event $\{Z(0)<\infty\}$. This implies that $\Bbb P[Z(0)=0$ or $+\infty] =1$. It follows that $$ \|X\|_2=\sqrt{\int_{\Bbb R} X(t)^2\,dt} = 0 \hbox{ or } +\infty,\quad\hbox{a.s.} $$ The non-triviality hypothesis $v>0$ implies that $P[\|X\|_2 =+\infty]>0$. This probability will therefore be $1$ if $X$ is ergodic, but it's easy to make (stationary, non-ergodic) examples in which $P[\|X\|=0]>0$.

Answer 2

Elaboration:

Stationary implies that $(X(t+s))_{s\ge 0}$ has the same distribution as $(X(s))_{s\ge 0}$, for each $t$. This is why $Z(t)$ and $Z(0)$ have the same distribution, being the same functional of $(X(t+s))_{s\ge 0}$ and $(X(s))_{s\ge 0}$, respectively. [The definition of stationarity involves the finite-dimensional distributions, with the implication that probabilities of events in $\mathcal X:=\sigma\{X(t): t\in\Bbb R\}$ are also shift invariant. That is, if we use $X^t$ to denote the time-shifted process $(X(t+s))_{s\ge 0}$ then $P[B] = P[\{\omega: X^t(\omega)\in B\}]$ for all $B\in\mathcal X$.]

A stationary process is ergodic provided the $\sigma$-field of invariant events is trivial (all such events have probability 0 or 1). The event $\{\|X\|_2=+\infty\}$ is invariant, so if it has positive probability then it must have probability 1. [Using the previous notation, an event $B$ is invariant provided $B=\{\omega: X^t(\omega)\in B\}$ for each $t$.]