By Birger Iversen

ISBN-10: 0387061371

ISBN-13: 9780387061375

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**Extra info for Ecole D'Ete de Probabilites: Processus Stochastiques**

**Sample text**

Since 1j; is subadditive and 1j;:S Q, we have 8 + U < 1j;(-f-l) :S 1j;(Un) + Q(-Sn) and consequently, we have {1j;(Un):S 8} ~ {Q( -Sn) > u} . Since 0"(8):S 1'£(8) :S P* (1j;(Un) :S 8) , we see that (2) holds. Suppose that E = -(1, ... ,1) . By the argument above with 8 + U replaced by 8 + t , we see that (2) holds. Suppose that E =I- ±(1, ... , 1) and let us define 7r:= {I :S i :S n I Ei = I} and v:= {1:S i:S n I Ei = -I}. Since E =I- ±(1, ... ,1), we see that 7r,V E II and since Xl"'" Xn are Fubini independent and 7r n v = 0 , we see that the pairs (Srr - f-lv, f-lv - Sv) and (Sv - f-l7r, f-l7r - S7r) satisfy the hypotheses of (D) in Thm.

E liJu5(hn)llu < 00 for any sequence {h n} E ll[H]. 2. 2). Let us also mention that the result is possibly new even for a Banach space X (this case is thoroughly examined in [1, 2]). V. Uglanov 5. ) : Y X X ----) Z is defined. (The X-hypocontinuity of the form means that, for any bounded set B c X and any neighborhood U E U(Z) , there exists a neighborhood V E U(Y) such that (V, B) c U. X-hypocontinuity follows from joint continuity in all variables and, as a rule, implies separate continuity.

The purpose of the paper is to show that the converse of the above assertion is valid for vector measures on a complete separable metric space with values in a certain semi-Montel space. Received by the editors December 1, 2002. 2000 Mathematics Subject Classification. Primary 28B05, 28C15; Secondary 46G 10. Key words and phrases. weak convergence of vector measures, uniform tightness, compactness criterion, semi-Montel space. Research supported by Grant-in-Aid for General Scientific Research No.

### Ecole D'Ete de Probabilites: Processus Stochastiques by Birger Iversen

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