By Van Der Merwe A. J., Du Plessis J. L.

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**Additional resources for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case**

**Example text**

X/ almost everywhere in G, then for an arbitrary x 2 G there exists a sequence xk # x as k ! xk /. By passing to the limit in the last equality as k ! 3. x/. tk ; k 2 N/, we see that h t ) h as t ! t/ ! x/; t ! t/ is regularly varying at inﬁnity. For any x 2 G, > 0, > 0 such that h is continuous at the points x and x, we see that Á h. b/ D h. 1 or h. x/ D Á˛ h. 21) where ˛ D ind r . We set D fyW y 2 G; y is a continuity point of hg. x / D lim h. x/ D ˛ lim h. x/ D ˛ #1; x2 "1; x2 h. x/; h. 2, h is continuous at x.

3, for any x; y > 0 as t ! 38) iD0 j D0 Therefore, for any " > 0, x > 0, y > 0 as t ! i; j / ! m; n/ does not increase in m. m; j /; j D0 where m D Œtx. 40) Since, as t ! t/ y = : 1 ; 50 1. t/ ! y = ; t ! t/ ! m; k/=n kD0 for t t0 and jj Œty/ Ä ıty C 1 where n D Œty. 1 The theorem is thus proved. 6. 8. m; n/ be monotone in m, and let ! m; i /=n iD0 as n ! n/. n/=n1C˛ / as n ! 1 and m n. The proof of this theorem repeats the above reasoning word for word. While studying branching processes, we will also use the following Tauberian theorem.

X/; t compact. D/ bjD1 j . k ! D/ t0 with cx D bjD . x/ is the density of the measure ˆ with respect to the Lebesgue measure in G. Let the set A , A 2 A. Then there exist " > 0 and > 0 such that "a < x < a for any x 2 A. It is easy to see, indeed, that if for arbitrary " > 0 there exists x 2 A such that x a" 2 E D Rn n G, then there exist sequences xk ! x 2 Ax and "k ! 0 as k ! 1 such that xk a"k 2 E. By passing to the limit as k ! 1, we ﬁnd that x 2 E because E is closed, which is impossible.