In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for highdimensional Orlicz balls. Roughly speaking, the latter shows that if V 1 and V 2 are Orlicz functions, then random points in the V 1 -Orlicz ball, conditioned on having a small V 2 -Orlicz radius, look like an appropriately scaled V 2 -Orlicz ball. In fact, we show that the limiting distribution in our Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V 2 -Orlicz radius.

We study the large deviation behavior of lacunary sums $$(S_n/n)_{n\in {\mathbb {N}}}$$ ( S n / n ) n ∈ N with $$S_n:= \sum _{k=1}^nf(a_kU)$$ S n : = ∑ k = 1 n f ( a k U ) , $$n\in {\mathbb {N}}$$ n ∈ N , where U is uniformly distributed on [0, 1], $$(a_k)_{k\in {\mathbb {N}}}$$ ( a k ) k ∈ N is an Hadamard gap sequence, and $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R is a 1-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed n and with a good rate function which is the same as in the case of independent and identically distributed random variables $$U_k$$ U k , $$k\in {\mathbb {N}}$$ k ∈ N , having uniform distribution on [0, 1]. When the lacunary sequence $$(a_k)_{k\in {\mathbb {N}}}$$ ( a k ) k ∈ N is a geometric progression, then we also obtain large deviation principles at speed n, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function f and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.

We study the large deviation behavior of lacunary sums (S n /n) n∈N with S n := n k=1 f (a k U ), n ∈ N, where U is uniformly distributed on [0, 1], (a k ) k∈N is an Hadamard gap sequence, and f : R → R is a 1-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed n and with a good rate function which is the same as in the case of independent and identically distributed random variables U k , k ∈ N, having uniform distribution on [0, 1]. When the lacunary sequence (a k ) k∈N is a geometric progression, then we also obtain large deviation principles at speed n, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function f and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.