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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 $f(0)+\cdots +f(k-1)\geq 0$?

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Edit: Please note that this answer does not solve the problem. It only shows that the attempt to reduce the problem to a geometric problem fails.

I think a have a counterexample to the geometric version with $k=3$. Let $$ v_1=(\sin a,\cos a,0) ,$$ $$ v_2=(0,\sin b,\cos b) ,$$ $$ v_3=(-\sin a,\cos a,0). $$ Then, $v_1\cdot \sum v_i = (\cos a)(\sin b+2\cos a) $, which can be made negative by choosing $a $ and $b $ adequately.

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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 $E [x_i|x_1=1]=3/8$.

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