Consider a random variable $X\sim_X(x)$ with $\operatorname{supp}{p_X}=[a,b]\subset{\mathbb R}$, and let $X_1, X_2, \ldots, X_n$ be $n$ $i.i.d.$ samples from $p_X(x)$, then we have their associated order statistics $X_{(1:n)}, X_{(2:n)}, \ldots, X_{(n:n)}$ such that $X_{(1:n)}\leqslant X_{(2:n)}\leqslant \cdots\leqslant X_{(n:n)}$.
Define the spacing between any two consecutive order statistics as $W_{(r:n)}=X_{(r+1:n)}-X_{(r:n)} (1\leqslant r\leqslant n-1)$, then in "Note on Francis Galton's problem" by K. Pearson (1902), it has been shown that
$$
{\mathbb E}\left[W_{(r:n)}\right]=\frac{n!}{(n-r)!r!}\int_{-\infty}^{\infty}F(x)^{n-r}[1-F(x)]^r \, dx, \hspace{2mm} 1\leqslant r\leqslant n-1,
$$
where $F(x)=\int_{-\infty}^xp_X(y) \, dy$ is the distribution function of $X$.
So, $\forall r, n$ such that $1\leqslant r\leqslant n-1$, $2\leqslant n$, do we have
$$
{\mathbb E}\left[W_{(r:n)}\right] \geqslant {\mathbb E}\left[W_{(r:n+1)}\right],
$$
i.e., the expected gap between any two consecutive order statistics will decrease or remain the same as $n$ increases?
Intuitively, this is correct since the more samples one draws, the denser the order statistics will be filled in the support. I also checked several specific distributions (e.g., uniform, Gaussian, etc), and the statement holds, but I have not figured out a good way to prove this for the general case. Hope someone can provide some inspiration.
Best Answer
This is not true. Consider a Bernoulli distribution with $p=\frac12$. (Technically, this doesn't have support on an interval, but we can approximate it arbitrarily closely with support on an interval.)
Then with $S_n=\sum_{k=1}^nX_n$ we have
$$ \mathsf E\left[W_{(r:n)}\right]=\mathsf P\left(X_{(r:n)}=0\land X_{(r+1:n)}=1\right)=\mathsf P(S_n=n-r)=2^{-n}\binom nr\;. $$
But
$$ 2^{-(n+1)}\binom{n+1}r\gt2^{-n}\binom nr $$
whenever
$$ \frac{n+1}{n-r+1}\gt2\;, $$
that is, for all $r\gt\frac{n+1}2$.
In this case, the support cannot be filled in; increasing $n$ merely changes the probability that these particular order statistics are separated by the constant gap. It could be that this effect can't occur if the distribution is unimodal.