  We determine the joint limiting distribution of adjacent spacings around a
central, intermediate, or an extreme order statistic $X_{k:n}$ of a random
sample of size $n$ from a continuous distribution $F$. For central and
intermediate cases, normalized spacings in the left and right neighborhoods are
asymptotically i.i.d. exponential random variables. The associated independent
Poisson arrival processes are independent of $X_{k:n}$. For an extreme
$X_{k:n}$, the asymptotic independence property of spacings fails for $F$ in
the domain of attraction of Fr\'{e}chet and Weibull ($\alpha \neq 1$)
distributions. This work also provides additional insight into the limiting
distribution for the number of observations around $X_{k:n}$ for all three
cases.
