# Counterexamples in Probability, 2nd Edition by Jordan M. Stoyanov By Jordan M. Stoyanov

Counterexamples (in the mathematical feel) are robust instruments of mathematical idea. This e-book covers counterexamples from chance conception and stochastic methods. This new increased version comprises many examples and the newest learn effects. the writer is thought of as one of many superior specialists within the box. includes numbers examples.

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Extra info for Counterexamples in Probability, 2nd Edition

Sample text

Then P(A t } = ~. A,A2 = [0, and independent. g. if Al U A similar example can be given in the discrete case. g. the sample space n = {I, 2, 3,4} with equaJJy likely outcomes and two classes A I and A2 where AI contains one of the outcomes of nand Az contains two of them. A simple calculation leads to a conclusion like that presented above. e. n E Ai and Ai. i = 1,2, is closed under intersection. 21 CLASSES OF RANDOM EVENTS AND PROBABILITIES SECTION 4. DIVERSE PROPERTIES OF RANDOM EVENTS AND THEIR PROBABILITIES Here we introduce and analyse some other properties of random events and probabilities.

See Renyi 1970). In this case it is usual to speak about mixing in the sense of ergodic theory (see Doukhan 1994). The mixing property can be extended as follows. The sequence {An} is called a stable sequence of events if for any B E ~ the following limit exists lim P(AnB) n--+oo = Q(B). According to Renyi (1970), Q is a measure on ~ which is absolutely continuous with respect to P. The Radon-Nikodym derivative dQ/dP = o:(w) exists and for every B E ~, Q(B) = a(w) dP. Here 0 ~ a(w) ~ 1 with probability 1.

In other words, there are numbers PI ,P2, ... ,Pk-I, all in (0, I), such that P(A j ) = PI for all j; P(AiAj) = P2 for all i < j; P(AiAj At) = P3 for all i < j < I etc. Like the independence property we can introduce the term exchangeability at level k for a fixed k meaning that P(A il ... A ik ) is the same for all choices of just k events from An regardless of what happens at levels higher than k, and lower than k. It turns out the collection An can be such that exchangeability property does not hold for others.