**Edit:** Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme. 


*Probabilistic version.*
Let $x=(x\_1,x\_2, \ldots) $ be an ergodic random sequence of bits. Let $k $ be a positive integer, and $i$ a random integer uniformly distributed in $\{1,\ldots, k\} $ independently of $x $. Is it alway the case that $Cov (x\_1,x\_i) \geq 0$?


Specifically, does the above hold when $x $ is a random rotation of a fixed periodic binary sequence?


*Quadratic programming version*.
For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x\_1,\ldots,x\_n)=\sum\_{i=1}^n\sum\_{j=0}^{k-1}x\_ix\_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is attained at $(1/n,\ldots,1/n)$? Where the unit simplex is the set $\{x\in\mathbb R^n:x\_i\geq 0\forall i,\ \sum x\_i=1\}$.


*Geometric version*.
Let $v\_1,\ldots,v\_k$ be vectors in a euclidian space. Suppose that the inner products satisfy $v\_i\cdot v\_j=f(|i-j|)$ for some function $f\colon\{0,\ldots,k-1\}\to\mathbb R$. Is it necessarily the case that $f(0)+\cdots +f(k-1)\geq 0$?



Correlation between the first and a random position of an ergodic bit sequence
The answer is "no". Here's a counterexample to the probabilistic version which translates to counterexamples to the other version. Consider the periodic binary sequence $a=(11000)^\omega $. Now, let $x $ be a random rotation of $a $. The expectation of $x\_i$ is $2/5$, which is greater than $E [x\_i|x\_1=1]=3/8$.


