By Dogeu D. (ed.), Lucaks E. (ed.), Rohatgi V.K. (ed.)

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One can see from the above approach that the key ingredient of the proof is to rewrite E{M//(W)} in terms of a functional of / ' . We formulate this in abstract form as follows. 1). 7) J\t\ we find t h a t fi(W + Ri+t)M(t)dt\ J < V(W < z + 50 + 5i) - $(z) + / J\t\

17) to obtain the following Berry-Esseen bound. 1) 24 Louis H. Y. Chen and Qi-Man Shao Proof: Write / = fz. 1) that E{/'(W« + t)}Ki(t) dt B{Wf(W)} = J2 "• poo = J2 / E{(WU +t)f{W® +t) + J t=i I{wW+tOO = X) / E { W / ( ^ ) ~ (W(i) + t)f{W«> + t)}Ki(t) dt. oo J^E / \Wf(W) - (H^W +t)/(W^w +t)\Ki{t)dt < X /""E/d^Wl + ^ / ^ d ^ l + ltl)}^*)* < (l + v ^ / 4 ) ^ / (E|6| + |t|)^(t), 2 since E{Ty^} < 1 and ^j and W^ are independent.

Consider a set of random variables {Xt,i G V} indexed by the vertices of a graph Q = (V, £). Q is said to be a dependency graph if, for any pair of disjoint sets Y\ and ^ in V such that no edge in £ has one endpoint in Fi and the other in F2, the sets of random variables {Xi,i G Fi} and {Xi,i G F2} are independent. Let D denote the maximal degree of G; that is, the maximal number of edges incident to a single vertex. Let Ai = {i} U {j S V: there is an edge connecting j and i} and Bi = \Jj£Ai Aj.

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