Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions

Abstract

For each t∈(−1,1), the exact value of the least upper bound H(t)=sup{E|X|3/E|X−t|3} over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment EX1/EX12=t, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M:=supL1(X)/L1(X−EX)=17+77/4, where L1(X)=E|X|3/(EX2)3/2 is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables X1,X2,… is proven in terms of the central Lyapunov ratio L1(X1−EX1) with the constant 0.3031·Ht(1−t2)3/2∈[0.3031,0.4517), t∈[0,1), which depends on the normalized first-moment t:=EX1/EX12 of random summands and being arbitrarily close to 0.3031 for small values of t, an almost 1.5 size improvement from the previously known one.