Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p ball in R n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to ∞, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p ∈ (1, ∞] (but not for p = 1), the corresponding rate function is "universal", in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the directions corresponding to the classical Cramér's theorem, and show that those directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of p balls.

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

The empirical mean of n independent and identically distributed (i.i.d.) random variables (X 1 , . . . , X n ) can be viewed as a suitably normalized scalar projection of the n-dimensional random vector X (n) .= (X 1 , . . . , X n ) in the direction of the unit vector n −1/2 (1, 1, . . . , 1) ∈ S n−1 . The large deviation principle (LDP) for such projections as n → ∞ is given by the classical Cramér's theorem. We prove an LDP for the sequence of normalized scalar projections of X (n) in the direction of a generic unit vector θ (n) ∈ S n−1 , as n → ∞. This LDP holds under fairly general conditions on the distribution of X 1 , and for "almost every" sequence of directions (θ (n) ) n∈N . The associated rate function is "universal" in the sense that it does not depend on the particular sequence of directions. Moreover, under mild additional conditions on the law of X 1 , we show that the universal rate function differs from the Cramér rate function, thus showing that the sequence of directions n

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