A [question][1] was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. Therefore, I am resurrecting the QA. Here is the question, in a bit brushed-up form: 

Let $(X_t)_{t\in\mathbb R}$ be a normalized [wide-sense stationary stochastic process][2] in $\mathbb R$, so that $g(t):=\text{Cov}(X_s,X_{s+t})$ does not depend on $s$ and $g(0)=1$. 
By [Bochner's theorem][3], $g$ must then be the [characteristic function][4] of a probability distribution. 

The question is: Is it possible that $\frac{1-g(t)}{t^2}\underset{t\to0}\longrightarrow0$? 

[1]: https://mathoverflow.net/questions/336659/existence-of-isotropic-random-field-with-slow-covariance/336662#336662

[2]: https://en.wikipedia.org/wiki/Stochastic_process#Stationarity 

[3]: https://en.wikipedia.org/wiki/Bochner%27s_theorem

[4]: https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)