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Extra info for A statistical method for the estimation of window-period risk of transfusion-transmitted HIV in dono

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These two classes of subsets of S form the smallest and the largest fields (a-field, monotone class). For any event A it is easy to check that the class A, consisting of the four events (0, A , A",S}, is closed under any operations: unions, intersections, and complements of members of A are again members of A. This class is an example of a field (a-field, monotone class) that contains the events A and A", and it is the smallest such field (u-field, monotone class). On the other hand, the class A, consisting of events {0, A , S}, is a monotone class, but neither a field nor a-field.

Increasing sequence) or when A1 3 A2 3 . . (decreasing sequence). 1 Zfthe sequence A1, Azl . . is increasing, then u m lim A, = n=l A,, and in case of a decreasing sequence, we have W lim A, = n=l A,. uzl uz*=, Proof. If the sequence is increasing, then the inner union ( A i ) in lirn sup A, remains the same independently of k so that lim sup A, = Ai. On the other hand, the inner intersection in lim inf A, equals Ak so that lim inf A, = uT=l Ak, which is the same as limsup A,, as was to be shown.

However, if s 2 + y 2 2 1/3, then ( 5 ,y) does not belong to any A, with even n. Thus lim sup A, # lim inf A,, and the sequence {A,} does not converge. + + 18 EXPERIMENTS, SAMPLE SPACES, A N 0 EVENTS Infinite operations on events play a very important role in the development of the theory, especially in determining limiting probabilities. The definitions below will prepare the ground for the considerations in the following chapters. In Chapter 2 we will introduce probability as a number assigned to an event.

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A statistical method for the estimation of window-period risk of transfusion-transmitted HIV in dono by Yasui Y.

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