A Maxwell principle for generalized Orlicz balls

Abstract

In [A dozen de Finetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist. 23(2) (1987), 397-423], Diaconis and Freedman studied the low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Rüschendorf and Naor and Romik unified these results by establishing a connection between ℓ N p balls and a p-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the formwhere φ : R → [0, ∞] is a function satisfying some fairly mild conditions; in particular, we cover the case of Orlicz functions. Our method is different from both Rachev-Rüschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cramér's theorem and the Gibbs conditioning principle, providing a natural framework beyond the pgeneralized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of ℓ N p balls. We find that there is a critical parameter t crit at which there is a phase transition in the behaviour of the low-dimensional projections: for t > t crit the coordinates of random vectors sampled from B N φ,t behave like uniform random variables, but for t ≤ t crit however the Gibbs conditioning principle comes into play, and here there is a parameter β t > 0 (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to e −β t φ(s) .

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

“…In 1991, Mogul'skiȋ [22] and Rachev and Rüschendorf [25] obtained an ℓ p analogue of the Poincaré-Maxwell-Borel lemma for the surface and cone measure respectively on a properly scaled ℓ p -sphere (note that surface and cone measure coincide whenever p ∈ {1, 2, ∞}). A simplification of the arguments can be found in the work of Naor and Romik [23,Theorems 3 and 4] and a significant generalization (including the case of Orlicz balls) has recently been obtained by Johnston and Prochno in [11]. In the ℓ p -versions of the Poincaré-Maxwell-Borel lemma, instead of the standard Gaussian distribution, the so-called p-generalized Gaussian distribution appears in the weak limit, the Lebesgue density being given by…”

“…Now take some R > 0 with R ≤ R. For the solution α(R) of Equation ( 30), we have that α(R) ≤ α(R). This holds since the map α → R V 2 (x)e αV 2 (x)−ϕ V 2 (α,0) d x is monotone increasing (see, e.g., p. 5 in [11]). Thus, if we are able to show that the function…”

“…Proposition 2.15. (Lemma C.1 in [8]) Let V be an Orlicz function and M V 1 (R) and d V be defined as in (11) and (12), respectively. Then, for a sequence…”

“…In order to get a nice geometric interpretation of Theorem B, we need the following limit theorem from [11]. We recall that λ (n,k) R,V is the distribution of the first k components of a random vector uniformly distributed on the Orlicz ball B n,V R , where R ∈ (0, ∞).…”

“…We recall that λ (n,k) R,V is the distribution of the first k components of a random vector uniformly distributed on the Orlicz ball B n,V R , where R ∈ (0, ∞). The distribution λ (n,k) R,V has a Lebesgue density on R k (see p.6 in [11]) which we shall denote by…”