By Olav Kallenberg

This is the 1st finished therapy of the 3 easy symmetries of chance thought - contractability, exchangeability, and rotatability - outlined as invariance in distribution below contractions, variations, and rotations. Originating with the pioneering paintings of de Finetti from the 1930's, the speculation has developed right into a precise physique of deep, appealing, and infrequently fabulous effects, comprising the fundamental representations and invariance homes in a single and a number of other dimensions, and showing a few unforeseen hyperlinks among a few of the symmetries in addition to to many different components of contemporary likelihood. such a lot chapters require just some simple, graduate point chance concept, and may be obtainable to any critical researchers and graduate scholars in likelihood and statistics. elements of the booklet can also be of curiosity to natural and utilized mathematicians in different components. The exposition is officially self-contained, with distinct references supplied for any deeper evidence from actual research or likelihood utilized in the ebook.

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**Example text**

I ∈ I, are separately exchangeable or contractable, if for any permutations or sub-sequences p1 , p2 , . . of N N we have d (ξ1 ◦ p1 , ξ2 ◦ p2 , . ) = (ξ1 , ξ2 , . ). 7 (separate exchangeability and independence, de Finetti) For any Borel space S and index set I, let ξi = (ξik ), i ∈ I, be inﬁnite, separately exchangeable sequences in S with directing random measures νi . Then the elements ξik are conditionally independent with distributions νi , given the family ν = {νi ; i ∈ I}. Proof: For convenience, we may write ξ = {ξi }, ξ \ ξi = {ξj ; j = i}, ξi \ ξik = {ξih ; h = k}.

N} denotes the uniform distribution on the set {1, . . , n}. L(ξ1 , . . , ξk ) − L(η1 , . . 10 (sampling equivalence) Let ξ1 , ξ2 , . . d. U{1, . . , n}, and let η1 , . . , ηn be the numbers 1, . . , n listed in their order of ﬁrst appearance in the sequence (ξj ). Then the ηj are exchangeable, and {ξj = ηj } ≥ 1 − P j≤k k(k − 1) , 2n 1 ≤ k ≤ n. (6) Proof: Fix any permutation p = (p1 , . . , pn ) of 1, . . , n. Then pη1 , . . , pηn lists the numbers 1, . . , n in their order of ﬁrst appearance in the sequence pξ1 , pξ2 , .

Nite and non-random. Put ξ¯ = ξ(S × ·), and let η = (η1 , . . , ηn ) be the associated sequence of marks. Then (i) ξ is contractable iﬀ ξ¯ and η are independent and contractable. In that case, (ii) ξ and η are simultaneously extreme, (iii) ξ and η are simultaneously exchangeable.