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andmlog 1 D log.1nSubstituting these expressions into (3.2), we obtainˇˇGn;m 2 pxp f C 1 2 log.1 p/ C 1 2 log 1p/ C log 1px: px pˇˇˇˇ 2 .1 C x=/D O; 2 .1 px=/for m n 1, where f .t/ WD t .1 t/ log.1 t/. This together with the inequalityˇˇf .t/ C t 2 2 C t 3 6 ˇ t 412.1 t/yieldsˇˇGn;mpx2q p C px22 C p2 x 36q 2p C 1 2 log.1 p2p/ˇˇˇˇ D O1 C x 2 C jxj pC px 4 :This completes the proof.Proof of theorems 1.1 and 1.3. Assume p 1=2 and take R D m=. Thenjı n;m j .1 C p.R 1//n .R 1/1/C e e.R 2R m R mD 2e R m=1 log u du : (3.3)Now we split the sum of d .˛/TV .L.X /; L.Y // into two partsXd .˛/TV .L.X /; L.Y // Djxj 1=6 p2=3 jı n;m j˛ CXjxj> 1=6 p2=3 jı n;m j˛:The first sum is evaluated by applying Lemma 3.1 and the second sum is estimated by inequality (3.3).These two estimates are enough to prove Theorem 2.1 when all p j ’s are equal. Once Theorem 2.1 isestablished, Theorems 1.1 and 1.3 will follow by exactly the same argument as we used above for Poissonbinomialdistribution.References[1] D. ALDOUS, Probability Approximations via the Poisson Clumping Heuristic, Springer-Verlag, NewYork, 1989.[2] A. D. BARBOUR, Asymptotic expansions in the Poisson limit theorem, Ann. Probab., 15 (1987), pp.748–766.[3] A. D. BARBOUR AND L. H.-Y. CHEN, Stein’s Method and Applications, Singapore University Pressand World Scientific Publishing Co., Singapore, 2005.[4] A. D. BARBOUR AND P. HALL, On the rate of Poisson convergence, Math. Proc. Cambridge Philos.Soc., 95 (1984), pp. 473–480.26