  Let $\left(a_{n}\right)_{n=1}^{\infty}$ be a lacunary sequence of positive
real numbers. Rudnick and Technau showed that for almost all
$\alpha\in\mathbb{R}$, the pair correlation of $\left(\alpha
a_{n}\right)_{n=1}^{\infty}$ mod 1 is Poissonian. We show that all higher
correlations and hence the nearest-neighbour spacing distribution are
Poissonian as well, thereby extending a result of Rudnick and Zaharescu to
real-valued sequences.
