Louis Sucheston

Stopping rules and tactics for processes indexed by a directed set

A new notion of tactic for processes indexed by a directed set is introduced. The main theorem, giving conditions under which tactics can be mapped on stopping times on the line, is applied to reduce some optimal stopping problems in the plane to the same problems on the line. In the case of independent random variables, one achieves a nearly complete reduction of the optimal reward problem to the linear case.

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||

On One-Parameter Proofs of Almost Sure Convergence of Multiparameter Processes

The martingale convergence theorem of R. Cairoli [-4] was proved assuming that the filtration satisfies a conditional independence assumption usualy called (F4). It belongs to the folklore of the subject (see also [-17]) that (F4) can be restated as a condition on commutation of conditional expectation operators. This formulation will allow us to derive Cairolis theorem from a simple general argument about operators on Orlicz spaces. The advantage is that one obtains a unified proof of multi-parameter versions of several other results: the theorems of Rota, Dunford-Schwartz, Akcoglu, and Stein. H. FNlmer was able to apply Proposition 2.1 below to random fields.

A refinement of the Riesz decomposition for amarts and semiamarts

A real-valued adapted sequence of random variables is an amart if and only if it can be written as a sum of a martingale and a sequence dominated in absolute value by a Doob potential, i.e., a positive supermartingale that converges to 0 in L1. The same holds for vector-valued uniform amarts with the norm replacing the absolute value.

Equal signs additive sequences in Banach spaces

Abstract A sequence ( e n ) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the e i s does not change when adjacent coefficients of equal sign are combined. Call the sequence ( e n ) regular if neither E nor its dual contain an isomorphic copy of c 0 . It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence ( e n ) is called IS or invariant under spreading if the norm of a linear combination of the e i s does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula ; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful.

On convergence of information in spaces with infinite invariant measure

SummaryThe paper extends the ergodic theorems of information theory (Shannon-MacMillan-Breiman theorems) to spaces with an infinite invariant measure. An L1 difference theorem and a pointwisa ratio theorem are proved, for the information of spreading partitions. For the validity of the theorems it is assumed that the supremum f* of the conditional information given the increasing “past” is integrable. Simple necessary and sufficient conditions for the integrability of f* are obtained in special cases: If the initial partition is composed of one state of a null-recurrent Markov chain, then f* is integrable if and only if the partition of this state according to the first return times has finite entropy.

On multiparameter ergodic and martingale theorems in infinite measure spaces

SummaryA unified proof is given of several ergodic and martingale theorems in infinite measure spaces.

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.

On mixing in infinite measure spaces

Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote σ-algebra (=tail σ-field). A transformation T is called mixing if (T−nA) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures π1,π2, \(\pi _1 ,\pi _2 ,\pi _1 \circ T^{ - n} - \pi _2 \circ T^{ - n} \to 0\) in total variation norm.