Discrete analog of the Orlicz-norm bounds for log-concave variables

Determine whether an Orlicz-norm sandwich inequality analogous to the continuous case holds for integer-valued log-concave random variables on the counting measure over Z. Specifically, establish whether for every Young function ψ and every log-concave random variable Y on Z with probability mass function f and maximum mass M(Y)=max_n f(n), there exist canonical discrete extremal distributions—such as a discrete uniform distribution on an integer interval and a one-sided geometric distribution—chosen to have the same maximum mass M(Y), for which the bounds ||U_d−E[U_d]||_ψ ≤ ||Y−E[Y]||_ψ ≤ ||Z_d−E[Z_d]||_ψ are valid.

Background

In the continuous setting (Lebesgue measure on R), the paper proves an Orlicz-norm ordering result (Theorem 2) for log-concave random variables X with fixed density maximum M(X): for every Young function ψ, the Orlicz norm of X−E[X] is sandwiched between that of a uniform distribution on an interval and that of an exponential distribution, both chosen to have the same maximum M.

When moving to the discrete setting (counting measure on Z), the authors indicate that their approach fails because asymmetric Laplace distributions are not equimeasurable in the discrete case, undermining a key step used in the continuous proof. Despite obtaining other discrete anticoncentration inequalities (including sharp bounds involving the geometric distribution as an extremal case for variance and fourth absolute central moment), the existence of a full Orlicz-norm analog remains unresolved.