Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and $$2=p_0<p_1<\ldots$$ be the increasing sequence of all primes. Let

$$ \forall_{n=1\ 2\ \ldots}\ \ d_n\, :=\, \frac{p_n-p_{n-1}}{p_{n+1}-p_{n-1}} $$

so that $\ p_n = (1-d_n)\cdot p_{n-1} + d_n\cdot p_{n+1}.\ $Consider arbitrary reals: $$0<a<b<1$$

**QUESTION**

Can you prove or disprove or provide a correct limit or that there is none or a reference for equality:

$$ \lim_{n=\infty}\ \frac 1n\cdot \left |\{k\in\mathbb J_n: a\le d_k\le b\}\right|\ =\ b-a $$

One may also consider the related questions about the frequency of the inequalities $\ p_n^2 > p_{n-1}\cdot p_{n+1}\ $ or $\ p_{n+1}\cdot p_{n+2} > p_n\cdot p_{n+3},\ $ or even some more precise questions too. (There is a slight bias toward the $\ >\ $ inequality because it holds when the intermediate primes or their arithmetic mean are close to the arithmetic mean of the external pair of primes; but this wouldn't matter in the limit, or would it?)