2023/05/30 by Michel Weber, Weber, Michel J. G.
Decision Sciences · #60F05 #60F15 #60G50 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2305.19372
openalex publication_date 2023/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let X=\Xj , j≥ 1\ be a sequence of independent, square integrable variables taking values in a common lattice \mathcal L(v 0,D )= \v k=v 0+D k , k∈ \Z\. Let Sn=X1+… +Xn, an= \mathbb E Sn, and \sn2=\rm Var(Sn)→ ∞ with n. Assume that for each j, \tXj =∑k∈ \Z\mathbb P\Xj=vk\\wedge\mathbb P\Xj=vk+1\>0. Using the Bernoulli part, we prove a general sharp correlation inequality extending the one we obtained in the i.i.d. case in \citeW3: Let 0<\tj≤ \tXj and assume that νn =∑j=1n \tj \uparrow ∞, n→ ∞. Let \kj∈ \mathcal L(jv0,D), j=1,2,… be a sequence of integers such that \rm(1) (κj-aj)/(\sj)=\mathcal O(1 ), \qq \rm(2) \sj \mathbb P\Sj=κj\ =\mathcal O(1). Then there exists a constant C such that for all 1≤ m