Resumen de Large deviations for random projections of ℓpℓp balls
Nina Gantert, Steven Soojin Kim, Kavita Ramanan
Let p∈[1,∞]p∈[1,∞]. Consider the projection of a uniform random vector from a suitably normalized ℓpℓp ball in RnRn 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 nn 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,∞]p∈(1,∞] (but not for p=1p=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 sequence of directions corresponding to the classical Cramér’s theorem, and show that those sequences of 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 ℓpℓp balls.
