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Edit: Since the geometric approach did not work, I phrase, now, 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 $f(0)+\cdots +f(k-1)\geq 0$?

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

Edit: The original question was probabilistic but it reduces to a question in euclidian geometry. I give the two versions.

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?

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

Edit: Since the geometric approach did not work, I phrase, now, 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 $f(0)+\cdots +f(k-1)\geq 0$?

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?

Edit: The original question was probabilistic but it reduces to a question in euclidian geometry. I give the two versions.

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?

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