Let *U*_{1}, *U*_{2}, . . . , *U*_{n–1} be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order *s* are defined as
$${G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}$$
with notation *U*_{0} = 0, *U*_{n} = 1, where
$${N^\prime=\left\lfloor n/s\right\rfloor}$$
is the integer part of *n*/*s*. Let
$${ N=\left\lceil n/s\right\rceil}$$
be the smallest integer greater than or equal to *n*/*s*, *f*_{m} (*u*), *m* = 1, 2, . . . , *N*, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic
$${f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}$$
is proved.