Classical works of Kac, Salem and Zygmund, and Erdos and Gal
have shown that lacunary trigonometric sums despite their dependency
structure behave in various ways like sums of independent and
identically distributed random variables. For instance, they satisfy a
central limit theorem and a law of the iterated logarithm. Those results
have only recently been complemented by large deviation principles by
Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan, showing that
interesting phenomena occur on the large deviation scale that are not
visible in the classical works. It turns out that the limit behaviour of
a lacunary trigonometric sum depends on number theoretic properties of
the underlying lacunary sequence, and may change drastically under small
perturbations of the lacunary sequence. This raises the question on what
scale such a phenomena kick in. At the beginning of the talk we will
introduce all the necessary probabilistic notation and discuss in
details historical and more recent results about limit theorems for
lacunary trigonometric sums. Then we move to the results regarding the
moderate deviation principles, i.e., limit theorems in a scale
interpolating between previously known central limit theorem and the
large deviation principle. Based on a joint work with Joscha Prochno.