The classical definition of martingales is extended to a more general case in the space of Banach lattices by V. Troitsky [6]. In the Banach lattice framework, martingales are defined without a probability space and the famous Doob’s convergence theorem was reproduced. Moreover, under certain conditions on the Banach lattice, it was shown that the set of bounded martingales forms a Banach lattice with respect to the point-wise order. In 2011, H. Gessesse and V. Troitsky [2] produced several sufficient conditions for the space of bounded martingales on a Banach lattice to be a Banach lattice itself. They also provided examples showing that the space of bounded martingales is not necessarily a vector lattice. Several other works have been done by other authors with regard to martingales in vector lattices, such as [4, 3].

In the theory of random processes, not just the study of martingale convergence is important, but the study of convergence of martingale-like stochastic sequences and processes, and the determination of interrelation between them are also crucial. So it is natural to ask if martingale-like sequences can be defined in a vector lattice or Banach lattice framework. In this article, we define and study martingale-like sequences in Banach lattices along the same lines as martingales are defined and studied in [6].

Classically, a martingale-like sequence is defined as follows (for instance, see a paper by A. Melnikov [5]). Consider a probability space (Ω,F,P) and a filtration (Fn)n=1∞ , i.e., an increasing sequence of complete sub-sigma-algebras of F . An integrable stochastic sequence x=(xn,Fn) is an L1 -martingale if limn→∞supm⩾nE|E(xm|Fn)−xn|=0. An integrable stochastic sequence x=(xn,Fn) is an E-martingale if P{ω:E(xn+1|Fn)≠xninfinitely often}=0.

Here we extend the definition of L1 -martingales and E-martingales in a general Banach lattice X following the same lines as the definition of martingales in Banach lattices in [6]. First we mention some terminology and definitions from the theory of Banach lattices for the reader convenience. For more detailed exploration, we refer the reader to [1]. A vector lattice is a vector space equipped with a lattice order relation, which is compatible with the linear structure. A Banach lattice is a vector lattice with a Banach norm which is monotone, i.e., 0⩽x⩽y implies ‖x‖⩽‖y‖ , and satisfies ‖x‖=‖|x|‖ for any two vectors x and y. A vector lattice is said to be order complete if every nonempty subset that is bounded above has a supremum. We say that a Banach lattice has order continuous norm if ‖xα‖→0 for every decreasing net (xα) with infxα=0 . A Banach lattice with order continuous norm is order complete. A sublattice Y of a vector lattice is called an (order) ideal if y∈Y and |x|⩽|y| imply x∈Y . An ideal Y is called a band if x=supαxα implies x∈Y for every positive increasing net (xα) in Y. Two elements x and y in a vector lattice are said to be disjoint whenever |x|∧|y|=0 holds. If J is a nonempty subset of a vector lattice, then its disjoint complement Jd is the set of all elements of the lattice, disjoint to every element of J. A band Y in a vector lattice X that satisfies X=Y⊗Yd is refered to as a projection band. Every band in an order complete vector lattice is a projection band. An operator T on a vector lattice X is positive if Tx⩾0 for every x⩾0 . A sequence of positive projections (En) on a vector lattice X is called a filtration if EnEm=En∧m . A sequence of positive contractive projections (En) on a normed lattice X is called a contractive filtration if EnEm=En∧m . A filtration (En) in a normed lattice X is called dense if Enx→x for each x in X. In many articles such as in [6], a martingale with respect to a filtration (En) in a vector lattice X is defined as a sequence (xn) in X such that Enxm=xn whenever m≥n .

Note that Definition 2 is equivalent to saying a sequence (xn) is an E -martingale if there exists l⩾1 such that Enxm=xn whenever m⩾n≥l . The symbol “ E ” stands for eventual so when we say (xn) is an E -martingale, we are saying that after a first few finite elements of the sequence, the sequence becomes a martingale.

Sequences defined by Definition 1 and Definition 2 are collectively called martingale-like sequences. Notice that every martingale (xn) in a vector lattice X with respect to a filtration (En) is obviously an E -martingale with respect to the filtration (En) . Moreover, every E -martingale (xn) in a Banach lattice X with respect to a contractive filtration (En) is an X-martingale with respect to the contrative filtration (En) . Note that for every x in a vector lattice X and a filtration (En) in X, the sequence (Enx) is an E -martingale with respect to the filtration (En) . If x is in a normed space X and (En) is a contractive filtration, then the sequence (Enx) is an X-martingale with respect to the contractive filtration (En) .

By considering any nonzero martingale (xn) in a Banach lattice X with respect to filtration (En) where x1 is nonzero without loss of generality, we can define a sequence (yn) such that y1=2x1 and yn=xn for all n⩾2 . Then one can see that (yn) is an E -martingale with respect to the filtration (En) . However, (yn) is not a martingale.

Note that every sequence which converges to zero is an X-martingale with respect to any contractive filtration (En) because if xn→0 and m>n n$]]> then ‖Enxm−xn‖⩽‖xm‖+‖xn‖→0 as n→∞ . So one can easily create an X-martingale (xn) which is not E -martingale by setting xn=1nx where x is a nonzero vector in X.

A martingale-like sequence A=(xn) with respect to a contractive filtration (En) on a normed lattice X is said to be bounded if its norm defined by ‖A‖=supn‖xn‖ is finite. Given a contractive filtration (En) on a normed lattice X, we denote the set of all bounded X-martingales with respect to the contractive filtration (En) by MX=MX(X,(En)) and the set of all bounded E -martingales with respect to the contractive filtration (En) by ME=ME(X,(En)) . With the introduction of the sup norm in these spaces, one can show that MX and ME are normed spaces. Keeping the notation M of [6] for all bounded martingales with respect to the contractive filtration (En) and from the preceding arguments, these spaces form a nested increasing sequence of linear subspaces M⊂ME⊂MX⊂ℓ∞(X) , with the norm being exactly the ℓ∞(X) norm.

The following proposition confirms that for any convergent element A=(xn) of MX we can find a sequence in ME that converges to A.

In [6] and [2] several sufficient conditions are established where the set of bounded martingales M is a Banach lattice. In [2], counter examples are provided where M is not a Banach lattice. So, one may similarly ask when are MX and ME Banach spaces and Banach lattices? We start by showing a counter example that illustrates that ME is not necessarily a Banach space. Example 5.

Let c0 be the set of sequences converging to zero. Consider the filtration (En) where En∑i=1∞αiei=∑i=1nαiei . Thus the sequence (yn) where yn=∑i=1n1iei is an E-martingale with respect to this filtration. We define a sequence of E-martingales Am as Am=(xnm) where xnm=∑i=n∞1iei,forn⩽m,yn/m,forn>m. m.\end{array}\right.\]]]> Define a sequence A=(xn) where xn=∑i=n∞1iei . We can see that A is not an E-martingale. But one can show that Am converges to A. Indeed, ‖Am−A‖=supn‖xnm−x‖=supn∈{m+1,m+2,…}‖yn/m− ∑i=n∞1iei‖→0 as m→∞.

Given a vector (Banach) lattice X and a filtration (respectively contractive) (En) on X, we can introduce order structure on the spaces ME and MX as follows. For two bounded E -martingales (respectively X-martingales) A=(xn) and B=(yn) , we write A⩾B if xn⩾yn for each n. With this order ME and MX are ordered vector spaces and the monotonicity of the norm follows from the monotonicity of the norm of X, i.e. for two E -martingales (respectively X-martingales) with 0⩽A⩽B , we have ‖A‖⩽‖B‖ . For two E -martingales (respectively X-martingales) A=(xn) and B=(yn) , one may guess that A∨B (or A∧B ) can be computed by the formulas A∨B=(xn∨yn) (or A∧B=(xn∧yn) ). We show in the following theorem that this is in fact the case in order for ME to be a vector lattice. However, this is not obvious to show in the case of MX .

Using the equivalence in Theorem 6, the following examples illustrate that ME is not always a vector lattice. Example 7.

Consider the classical martingale (xn) in L1[0,1] where xn=2n1[0,2−n]−1 with the filtration (Fn) where Fn is the smallest sigma algebra generated by the set {[0,2−n],(2−n,2−n+1],…,(1−2−n,1]}. One can easily show that En|xn+1|=E[|xn+1||xn]≠|xn| for every n and the sequence (|xn|) fails to be an E -martingale. Hence, Theorem 6 implies that ME is not a vector lattice.

Under the pointwise order structure on MX , for an X-martingale A=(xn) , we can refer to Example 8 to show that the sequence (|xn|) is not necessarily an X-martingale. However, under certain assumptions, we can show that (|xn|) is an X-martingale for every X-martingale A=(xn) making MX a Banach lattice.