# Existence of stationary stochastic processes with very high correlation

A question 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 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, $$g$$ must then be the characteristic function of a probability distribution.

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

$$\newcommand{\R}{\mathbb{R}}$$ This is only possible when $$g=1$$ identically. Indeed, otherwise, by Hoeffding's Lemma 3.2, page 217, one of the following two alternatives must take place:
(i) $$g$$ is the characteristic function of a degenerate distribution, so that $$g(t)=e^{itx}$$ for some real $$x\ne0$$ and all real $$t$$, and hence $$\frac{1-g(t)}{t^2}\underset{t\to0}\longrightarrow0$$ is false;
(ii) there are positive real numbers $$b$$ and $$c$$ such that $$1-|g(t)|\ge ct^2$$ for $$|t|\le b$$, which again precludes $$\frac{1-g(t)}{t^2}\underset{t\to0}\longrightarrow0$$, because $$|1-g(t)|\ge1-|g(t)|$$.