A Gaussian process with $E(X_t-X_0)^2\sim\vert \log t\vert^{-\beta}$

Question

Exercise 6.1 (Section 4.5) in: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes by R. Adler starts:

Let $X_t$ be a stationary Gaussian process on $[0,1]$ and assume $E(X_t-X_0)^2 \sim \vert \log t\vert^{-\beta}$, $\beta>0$.

I have two questions

1. Do you know what does the $\sim$ symbol means? My guess is that this means that $E(X_t-X_0)^2$ can be upper and lower bounded by positive constants times $\vert \log t\vert^{-\beta}$

2. Why such a Gaussian process exists? Of course I know this is related with the positive definiteness of the covariance functions.

Answer 1

1. As mentioned in the comments, this should mean that $E(X_t-X_0)^2/\vert \log t\vert^{-\beta}\to 1$, but, of course, as $t\to 0$, not $\infty$ or $1$.

2. A stationary Gaussian process can be constructed by means of a moving average representation: $$ X_t = \int_{-\infty}^t f(t-s) dW_s $$ with $W$ being a standard Wiener process and some $f\in L^2(0,\infty)$. Then one needs to find $f$ such that $$E(X_t-X_0)^2 = \int_0^t f(u)^2 du + \int_{0}^\infty |f(u) - f(u+t)|^2 dt \sim \vert \log t\vert^{-\beta}, t\to 0+.$$ This is an easy exercise, but please write if you will have any problems with this.