A Complete Moment Convergence Theorem for Extended Negatively Dependent Random Variables Under Slowly Varying Weights

We prove a complete moment convergence criterion for weighted maximal partial sums of extended negatively dependent (END) random variables under slowly varying weights. For every r>1, and for triangular weight arrays that are uniformly bounded, quadratically non-degenerate, and uniformly non-degenerate on their active coefficients, we show that the summability of nr−1l(n)E[(Sn*/n−ε)+] for all ε>0 is equivalent to the weighted moment condition E[|X|r+1l(|X|)]<∞. The slowly varying factor l gives a refined borderline scale: it weakens the pure (r+1)-moment condition when l(t)→0, strengthens it when l(t)→∞, and recovers the classical scale when l is bounded away from zero and infinity. The proof uses weight-dependent monotone clipping, a Rosenthal-type maximal inequality for END sequences, Potter bounds and Karamata-type estimates for slowly varying functions, and a Bonferroni lower-bound argument based on a linear set of significant coefficients. Particular attention is paid to the preservation of the END structure under clipping, centering, and signed weights. Several corollaries and borderline heavy-tail examples are included, and possible modeling interpretations are briefly discussed without claiming finite-sample risk bounds beyond the theorem.