G. A. Edgar

Amarts: A class of asymptotic martingales a. Discrete parameter

A sequence (Xn) of random variables adapted to an ascending (asc.) sequence n of [sigma]-algebras is an amart iff EX[tau] converges as [tau] runs over the set T of bounded stopping times. An analogous definition is given for a descending (desc.) sequence n. A systematic treatment of amarts is given. Some results are: Martingales and quasimartingales are amarts. Supremum and infimum of two amarts are amarts (in the asc. case assuming L1-boundedness). A desc. amart and an asc. L1-bounded amart converge a.e. (Theorem 2.3; only the desc. case is new). In the desc. case, an adapted sequence such that (EX[tau])[tau][set membership, variant]T is bounded is uniformly integrable (Theorem 2.9). If Xn is an amart such that supnE(Xn - Xn-1)2 0 in L1. Then Zn --> 0 a.e. and Z[tau] is uniformly integrable (Theorem 3.2). If Xn is an asc. amart, [tau]k a sequence of bounded stopping times, k a.e. on G and lim inf Xn = -[infinity], lim sup Xn = +[infinity] on Gc (Theorem 2.7). Let E be a Banach space with the Radon-Nikodym property and separable dual. In the definition of an E-valued amart, Pettis integral is used. A desc. amart converges a.e. on the set {lim sup ||Xn||

Pointwise convergence in terms of expectations

This paper is concerned with the connection between almost sure convergence of a sequence of random variables and convergence of certain related expectations. Theorems of the kind we are interested in were proved by Meyer I-7, p. 232] and Mertens [6, p. 47-1 in the continuous-parameter case, and by Baxter 1-11 in the discrete-parameter case. For example, Baxters theorem is the following: Let (X~)~_>_I be a sequence of random variables with values in a compact metric space S, and let the set F of bounded stopping times be directed by the obvious ordering. Then (X~)n_>l converges almost surely if and only if the generalized sequence (6 ~b (X~))~r of expectations converges for every real-valued continuous function q5 on S. In the present paper we generalize this theorem in two ways: we replace S by an arbitrary complete separable metric space, and we use as few test functions ~b as possible. IfS is the real line, the single test function ~b (x) = x suffices (Theorem 2); for any complete separable metric space, a countable set of functions suffices (Theorem 3); and for a separable Banach space, there is a countable set of convex functions which suffices (Theorem 4). We have included a different proof of the key step in Baxters proof (Corollary 1), in order to make the present paper selfcontained.

Borel subrings of the reals

A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positive Hausdorff dimension is equal to either R or C.

Extremal integral representations

Abstract Let X be a closed bounded convex subset with the Radon-Nikodym property of a Banach space. For tight Borel probability measures μ, v on X , define μ ≺ v iff there is a dilation T on X such that T ( μ ) = v . Then, for every x ϵ X , there is a measure μ on X which is maximal in the partial order ≺ and which has barycenter x . If X is separable, then μ (ex X ) = 1 for all maximal measures μ. In general, a maximal measure need not be “on” ex X in this strong sense. If X is weakly compact, then a maximal measure is “on” ex X in the looser sense that μ ( B ) = 1 for all weak Baire sets B ⊇ ex X .

A cartesian closed category for topology

Abstract By replacing sequences by sets in definitions of Frechet and Urysohn, one obtains the definition of an L ∗ space. The category of L ∗ spaces properly contains the category of topological spaces, it is cartesian closed, and it has other properties which make it convenient for topology. The category of L ∗ spaces is properly smaller than the category of convergence spaces (Limesraame) of Cook and Fischer, but properly larger than the category of epitopological spaces of Antoine.

Noncompact simplexes in Banach spaces with the Radon-Nikodým property

Abstract It is well known that a compact convex subset C of a locally convex topological vector space is a simplex if and only if each point x of C admits a unique probability measure on the extreme points of C with barycenter x . An exact analog of this result is proved for a closed and bounded separable convex subset of a Banach space with the Radon-Nikodým Property, and a weaker analog is proved in the nonseparable case.

A fractal puzzle

Figure 2 shows the 14 different cards that are (potentially) to be used in the puzzle. Each one is an equilateral triangle with side 5 cm. The back of each card shows the mirror image of the front. There are in fact several versions of the puzzle. The trivial version of the puzzle uses six copies of card A. They are to be assembled into a picture of the Barnsley wreath that is 10 cm wide. The elementary version of the puzzle uses 24 cards (12 copies of card A, 6 copies of card B, and 6 blank cards Z). They are to be assembled into a picture of the wreath that is 20 cm wide. The novice version of the puzzle uses 96 cards. (Each version has four times the number of cards as

Asplund operators and A. E. convergence

. Let E and F be Banach spaces, and U: E --> F a bounded linear operator. The following are equivalent: 1. o2. (a) U is an Asplund operator.3. (b) Let (Xn) be a sequence of Bochner measurable functions with values in E and supn||;Xn||

On vector-valued amarts and dimension of banach spaces

If (Xn)n~ N is an amart of class (B) taking values in a Banach space with the Radon-Nikodym property, then Xn converges weakly a.s., as proved in [4]. Examples exist in [4] and [7] which show that strong convergence may fail, but recently Alexandra Bellow [2] proved the following result: A Banach space E is finite-dimensional if (and only if) every E-valued amart of class (B) converges strongly a.s. We prove here that if p is fixed, 1 < p < o% then a Banach space E is finite-dimensional if (and only if) every LV-bounded E-valued amart converges weakly a.s. The point of this is that in the amart convergence theorem for an infinitedimensional Banach space, the assumption (B) cannot be weakened any more than the conclusion that weak a.s. convergence holds can be strengthened. Let (E2,~,P) be a probability space, N = { 1 , 2 . . . . }, and let (~),EN be an increasing sequence of a-algebras contained in ~ . A stopping time is a mapping r: s w { oo }, such that {7 = n} ~o~ for all n EN. The collection of bounded stopping times is denoted by T; under the natural ordering Tis a directed set. (The notation and the terminology of the present note are close to those of our longer article [7].) Let E be a Banach space and consider a sequence (X, ) ,~ of E-valued random variables adapted to (~),~N, i.e. such that X~: s is d -s t rongly measurable. We will write E X (expectation of X) for the Pettis integral [9] of the random variable X. The sequence (X,) is called an amart iff each X, is Pettis integrable and limr r EXr exists in the strong topology of E. An adapted sequence (Xn) is said to be of class (B) iff

Amarts: A class of asymptotic martingales B. Continuous parameter

A continuous-parameter ascending amart is a stochastic process (Xt)t[set membership, variant]+ such that E[X[tau]n] converges for every ascending sequence ([tau]n) of optional times taking finitely many values. A descending amart is a process (Xt)t[set membership, variant]+ such that E[X[tau]n] converges for every descending sequence ([tau]n), and an amart is a process which is both an ascending amart and a descending amart. Amarts include martingales and quasimartingales. The theory of continuous-parameter amarts parallels the theory of continuous-parameter martingales. For example, an amart has a modification every trajectory of which has right and left limits (in the ascending case, if it satisfies a mild boundedness condition). If an amart is right continuous in probability, then it has a modification every trajectory of which is right continuous. The Riesz and Doob-Meyer decomposition theorems are proved by applying the corresponding discrete-parameter decompositions. The Doob-Meyer decomposition theorem applies to general processes and generalizes the known Doob decompositions for continuous-parameter quasimartingales, submartingales, and supermartingales. A hyperamart is a process (Xt) such that E[X[tau]n] converges for any monotone sequence ([tau]n) of bounded optional times, possibly not having finitely many values. Stronger limit theorems are available for hyperamarts. For example: A hyperamart (which satisfies mild regularity and boundedness conditions) is indistinguishable from a process all of whose trajectories have right and left limits.