By D. Kannan

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Extra info for An introduction to stochastic processes

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The determination of such x or y proceeds via a sequence of nested sets An,in say for ∞ which N (An,in ) ≥ N (An+1,in+1 ) ≥ 1 for all n with n=1 An,in = {x} say. It follows that an enumeration of the points of N is thereby determined, for a sequence y1 , y2 , . . , within a ﬁnite number of steps for each yr even before its precise location is known. Now write yr = lim j Aj,ij,r for some monotonic decreasing sequence of sets Aj,ij,r for which N (Aj,ij,r ) = 1 for all suﬃciently large j. For any given ﬁnite enumeration of points, the position in the enumeration is found from a ﬁnite number of elements of dissecting systems, with all the associated counting measures N (·) measurable.

Ak ; x1 , . . , xk ) = Fk (Ai1 , . . , Aik ; xi1 , . . , xik ). (b) Consistency of marginals. For all k ≥ 1, Fk+1 (A1 , . . , Ak , Ak+1 ; x1 , . . , xk , ∞) = Fk (A1 , . . , Ak ; x1 , . . , xk ). The ﬁrst of these conditions is a notational requirement: it reﬂects the fact that the quantity Fk (A1 , . . , Ak ; x1 , . . , xk ) measures the probability of an event {ω: ξ(Ai ) ≤ xi (i = 1, . . , k)}, that is independent of the order in which the random variables are written down. The second embodies an essential requirement: it must be satisﬁed if there is to exist a single probability space Ω on which the random variables can be jointly deﬁned.

18) (with Pr replaced by P ). s. s. and, being the limit of an integer-valued sequence, is itself integervalued or inﬁnite. 18b), we have P {ζn (A) = 0} = ψ(A) for all n, so P {N (A) = 0} = ψ(A) (all bounded A ∈ R). s. 18) (with P and ψ replacing P and P0 ), reduces to condition (iii). , n→∞ n→∞ and thus N is ﬁnitely additive on R. Let {Ai } be any disjoint sequence in R with bounded union ∞ Ai ∈ R; A≡ i=1 ∞ i=1 ∞ N (Ai ). s. Deﬁne events Cr ∈ E for r = 0, 1, . . by C0 = {N : N (A) = 0} and Cr = {N : N (Br ) = 0 < N (Br−1 )}.