SUBDIAGONAL AND ALMOST UNIFORM DISTRIBUTIONS

A distribution (function) F on [0 ; 1] with F ( t ) less or equal to t for all t is called subdiagonal . The extreme subdiagonal distributions are identi(cid:12)ed as those whose distribution functions are almost surely the identity, or equivalently for which F (cid:14) F = F . There exists a close connection to exchangeable random orders on f 1 ; 2 ; 3 ; : : : g . In connection with the characterization of exchangeable random total orders on N an interesting class of probability distributions on [0 ; 1] arizes, the socalled almost uniform distributions, de(cid:12)ned as those w 2 M 1+ ([0 ; 1]) for which w ( f t 2 [0 ; 1] j w ([0 ; t ]) = t g ) = 1 ; i.e. the distribution function F of w is w {a.s. the identity. The space W of all almost uniform distributions parametrizes in a canonical way the extreme exchangeable random total orders on N ; as shown in [1]. If (cid:23) is any probability measure on R with distribution function G ; then the image measure (cid:23) G is almost uniform, see Lemma 3 in [1]. In this paper we show another interesting \extreme" property of W : calling (cid:22) 2 M 1+ ([0 ; 1]) subdiagonal if (cid:22) ([0 ; t ]) (cid:20) t for all t 2 [0 ; 1] ; we prove that the compact and

In connection with the characterization of exchangeable random total orders on N an interesting class of probability distributions on [0, 1] arizes, the socalled almost uniform distributions, defined as those w ∈ M 1 + ([0, 1]) for which w({t ∈ [0, 1]|w([0, t]) = t}) = 1, i.e. the distribution function F of w is w-a.s. the identity. The space W of all almost uniform distributions parametrizes in a canonical way the extreme exchangeable random total orders on N , as shown in [1]. If ν is any probability measure on R with distribution function G , then the image measure ν G is almost uniform, see Lemma 3 in [1]. In this paper we show another interesting "extreme" property of W : , we prove that the compact and convex set K of all subdiagonal distributions on [0, 1] has precisely the almost uniform distributions as extreme points. A simple example shows that K is not a simplex.

Lemma. Let a < b, c < d and
Then C is compact and convex (w.r. to the pointwise topology) and Proof. If ϕ([a, b]) = {c, d} then ϕ is obviously an extreme point. Suppose now that ϕ ∈ ex(C). We begin with the simple statement that on [0, 1] all functions f α (x) := x + α(x − x 2 ) , for |α| ≤ 1 , are strictly increasing from 0 to 1. If ϕ ∈ C then ψ := (ϕ − c)/(d − c) increases on [a, b] from 0 to 1, hence ψ α := f α • ψ has the same property. So ϕ α := (d − c)ψ α + c increases from c to d , i.e. ϕ α ∈ C for |α| ≤ 1 ; note that ϕ = ϕ 0 . Now ψ = 1 2 (ψ α + ψ −α ) and ϕ = 1 2 (ϕ α + ϕ −α ) which shows that ϕ is not extreme if ϕ = ϕ α . We note the equivalences (for α = 0) Both the classes of subdiagonal as well as almost uniform distributions being defined via their distribution functions, we will now work directly with these and consider K as those distribution functions F on [0, 1] for which F ≤ id. Theorem 2 in [1] can then be reformulated as The announced result is the following: Proof. "⊇": Let F ∈ W, G, H ∈ K such that F = 1 2 (G + H) . We now make use of the particular "shape" of almost uniform distribution functions: either t is a "diagonal point" of F , i.e. F (t) = t , or t is contained in a "flat" of F , i.e. in an interval ]a, b[ on which F has the constant value a , cf. Lemma 2 in [1]. If F (t) = t then certainly , implying 0 < s < 1 and F (s) < s . We may and do assume that F (t) < F (s) for all t < s , otherwise with s 0 := inf{t < s|F (t) = F (s)} we would still have We shall first consider the case that F is constant in a right neighbourhood of s , i.e. for some v ∈]s, 1] we have F |[s, v[ ≡ F (s) , and again we may and do assume that v is maximal with this property, i.e. F (v) > F (s) .
If F (s−) < F (s) , then for sufficiently small ε > 0 then G α is a distribution function for |α| ≤ 1 . Since F = 1 2 (G α + G −α ) we are done once we know that G α is subdiagonal for sufficiently small |α| . For this to hold we only need to know that cf. Remark 2. Now by right continuity there is some Together this gives ( * ) .
It remains to consider the case so that another application of the Lemma shows F to be not extreme in K . 2 In order to see that K is not a simplex, consider the following four almost uniform distribution functions F 1 , ...F 4 , determined by their resp. set of diagonal points D 1 , ..., D 4 : Let us shortly describe the connection of the above theorem to exchangeable random orders.
and such that either (j, k) or (k, j) ∈ V for all j, k ∈ N . The set V of all total orders is compact and metrisable in its natural topology, and a probability measure µ on V is called exchangeable if it is invariant under the canonical action of all finite permutations of N (see [1] for a more detailed description). A particular class of such measures arises in this way: let X 1 , X 2 , ... be an iid-sequence with a distribution w ∈ W .
This defines (uniquely) an exchangeable random total order, and the main result in [1] shows that {µ w |w ∈ W} is the extreme boundary of the compact and convex set of all exchangeable random total orders (on N ), which furthermore is a simplex. Now, given some exchangeable random total order µ , there is a unique probability measure ν on W such that which in a way is the " first moment measure" of ν .
One might believe that only very " simple" probability values depend on ν via ν , but in fact, due to the defining property of almost uniform distributions, also many " higher order" probabilities have this property. For example where X 1 , X 2 : [0, 1] 2 −→ [0, 1] denote the two projections.