New characterizations of the S topology on the Skorokhod space

The $S$ topology on the Skorokhod space was introduced by the author in 1997 and since then it proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the $S$ topology. It is shown that the convergence of sequences in the $S$ topology admits a compact description, exhibiting the locally convex character of the $S$ topology. It is also shown that $S$ is, up to some technicalities, finer than any linear topology which is coarser than Skorokhod's $J_1$ topology. The paper contains also definitions of extensions of the $S$ topology to the Skorokhod space of functions defined on $[0,+\infty)$ and with multidimensional values.


Introduction
The S topology on the Skorokhod space D = D([0, T ]) of càdlàg functions has emerged as a result of a chain of observations made in eighties and nineties of the twentieth century.
In 1984 Meyer and Zheng [29] considered certain conditions on truncated conditional variations of stochastic processes and proved that these conditions give uniform tightness of the processes in some topology (nowadays called the Meyer-Zheng topology) on D.
A year later Stricker [36] proved that it is possible to relax the Meyer-Zheng conditions to uniform tightness of random variables { X n ∞ } and {N a,b (X n )}, for each pair of levels a < b (where N a,b (x) is the number of up-crossings of levels a < b by function x).
Stricker still operated with the Meyer-Zheng topology. It was clear for Kurtz [20] that such conditions give much more. But an ad hoc device constructed by Kurtz did not have a topological character.
The S topology was constructed by the author in [13]. This step was final in the sense that Stricker's conditions are equivalent to the uniform tightness in the S topology.
It should be emphasized, that the S topology, as considered in [13], is sequential and it is still not known if it is completely regular. So at the moment of its creation there was no formalism to deal with it within the Probability Theory. Such formalism has been provided in [14]. And an efficient tool -the almost sure Skorokhod representation in non-metric spaces, given in [12] -made the S topology an operational and useful device in many problems.
In the present paper we provide some complementary information related to the S topology. Section 2 restates the definition and basic properties of the S topology. The S topology was defined in [13] by means of so-called L-convergence −→ S , which leads to the topological convergence * −→ S via the Kantorovich-Vulih-Pinsker-Kisynski recipe (we refer to Section 6 for a primer on sequential spaces).
In Section 3 we provide a closed form formula for the * −→ S convergence. In Section 4 we find a position for S in the hierarchy of topologies on D, by showing that it is essentially finer than any linear topology which is coarser than the Skorokhod J 1 topology.
In Section 5 we extend the notion of the S topology to the case of infinite time horizon and to functions with multidimensional values.

Definition of the S topology
All results in this section are taken from [13]. We need some standard notation first. 2. D is naturally equipped with the sup-norm x ∞ = sup t∈[0,T ] |x(t)|.
3. For a < b, N a,b (x) is the number of up-crossings of levels a and b by function x ∈ D. In other words, N a,b (x) is the largest integer k such that there are numbers where the supremum is taken over all 0 = t 0 < t 1 < . . . < t m = T, m ∈ N.
The S topology is defined in terms of S-convergence. Definition 2.1 (S-convergence). We shall write x n −→ S x 0 if for every ε > 0 one can find elements v n,ε ∈ V, n = 0, 1, 2, . . . which are ε-uniformly close to x n 's and weakly- * convergent: x n − v n,ε ∞ ≤ ε, n = 0, 1, 2, . . . , for each continuous function f : Moreover, by the Banach-Steinhaus theorem, relation (2.2) implies sup n v n,ε (T ) < +∞. (2.5) For the sake of brevity of formulation of the next theorem let us list some conditions describing properties of a subset K ⊂ D.
(iii) For every ε > 0 and every x ∈ K there exists v x,ε ∈ V such that and sup x∈K v x,ε < +∞.
(2.10) Definition 2.4 (The S topology). The S topology is the sequential topology on D generated by the L-convergence −→ S . Remark 2.5. " −→ S " is not an L * -convergence. In the next section we shall give a closed form of " Characterizations of the S topology Let A n ∈ A, n = 0, 1, 2, . . .. We will say that    x 0 (t) dA 0 (t), Proof. Suppose that x n * −→ S x 0 . Then x n (T ) → x 0 (T ) is a consequence of (2.4) and property (2.1) of functions {v n,ε }. In order to prove (3.3) one could observe that this convergence is a very particular (deterministic) case of Theorem 1 (or Theorem 5) in [11]. But it is more instructive to give here a direct proof.
A 0 (t) − A n (t) dv n,ε (t) = I 1 (n) + I 2 (n) + I 3 (n). Now let us assume that x n (T ) → x 0 (T ) and (3.3) holds for every sequence {A n } ⊂ A, A n −→ τ A 0 . We claim that it is enough to establish relative S-compactness of {x n }. Indeed, then in every subsequence {n } we can find a further subsequence {n } such that x n −→ S y 0 , for some y 0 ∈ D. Take a function f ∈ L 1 ([0, T ]) and define A f (t) = t 0 f (u) du. Then A f ∈ A and by the first part of the proof, Hence for each integrable f we have and, consequently, y 0 = x 0 almost everywhere. Since they are càdlàg functions, y 0 = x 0 on [0, T ). And y 0 (T ) = x 0 (T ) holds by our assumption.
So far we have proved that in every subsequence {n } we can find a further subsequence {n } along which x n −→ S x 0 . Hence, by the KVPK recipe, x n * −→ S x 0 . In order to prove the relative S-compactness of {x n } it is necessary to adjust integrands A n in a way suitable for the particular functional determining the relative S-compactness via Theorem 2.3.
First let us consider condition (2.6). Suppose that sup n x n ∞ = +∞. Then there exists a subsequence n k and numbers t k ∈ [0, T ) such that a k = |x n k (t k )| → ∞. Without loss of generality we may assume that a k = x n k (t k ) and t k < T . By the right-continuity of x n k we can find numbers h k such that t k + h k < T and It follows that (3.3) cannot be satisfied. In order to cope efficiently with condition (2.7) we need the following lemma.
Proof. Let

Then we have for
Now suppose that condition (2.7) does not hold for K = {x n }. This means that for some a < b and along some subsequence {n } Without loss of generality we may assume that N a,b (x n ) ≥ 2. For each n , let A n = A f n be given by Lemma 3.4

. Then
Proof. Clearly, (3.8) implies (3.3). To prove the converse, assume (3.3) and suppose that (3.8) does not hold for some relatively τ -compact set A ⊂ A. This means that for some η > 0 there exists a subsequence {n } and elements A n of A such that for all n Passing to a τ -convergent subsequence A n we obtain a contradiction with (3.3).
Remark 3.6. Denote by Σ the locally convex topology on D given by the seminorm ρ 1 (x) = |x(1)| and the seminorms where A runs over relatively τ -compact subsets of A. Then x n * −→ S x 0 if, and only if, x n −→ Σ x 0 and so S ⊃ Σ, for S is sequential.  optimal transport on the Skorokhod space. It seems that in problems of such type the local convexity related to Σ can be a useful tool as well.

S in the hierarchy of topologies
Let us begin with listing some facts on topologies on D.
1. D with norm · ∞ is a Banach space, but non-separable. 2. The Skorokhod J 1 topology is metric separable and D, J 1 is topologically complete. For definition and properties of J 1 we refer to Billingsley's classic book [6] rather than to its second edition.
3. It is easy to show that x n −→ J1 x 0 implies x n −→ S x 0 , hence the S topology is coarser than J 1 . 4. It was shown in [4] that the S topology is coarser than Skorokhod's M 1 topology (see [33] for definitions of four Skorokhod's topologies).
5. S is incomparable with Skorokhod's M 2 topology! 6. D, J 1 is not a linear topological space, for addition is not sequentially J 1 -continuous, as Figure 1 shows. 7. On the contrary, the sequence {f n } defined in Figure 1 is S-convergent to 0 and exhibits a typical for S phenomenon of self-cancelling oscillations. Addition is sequentially continuous in S! 8. We do not know whether addition is continuos, as a function on the product D × D with the product topology S × S (in general sequential continuity does not imply continuity). Therefore we do not know, whether (D, S) is a linear topological space.  σ is coarser than the uniform topology generated by the norm · ∞ .  Then σ is coarser than the S topology.
Proof. We claim that it is enough to prove that x n −→ S x 0 implies x n −→ σ x 0 along some subsequence {n }. Indeed, this implies that any σ s -closed set is also S-closed (for S is sequential), and so σ s is coarser than S. Since we always have σ ⊂ σ s , our claim follows.
So let us assume that x n −→ S x 0 . For ε > 0 and x ∈ D let us define  Since v ε (x) varies only through M ε (x) jumps, it can be represented as a sum of M ε (x) + 1
By (4.4) and (4.5) we obtain It follows that the sequence {v ε (x n )} lives in the algebraic sum is linear K ε is relatively σ-compact as well. This means that in every subsequence {n } one can find a further subsequence {n } such that v ε (x n ) −→ σ v ε , for some v ε ∈ D. But we can say more: by the special form (4.6) of elements v ε (x n ) (bounded number of jumps with bounded amplitudes) we may extract a further subsequence {n } such that v ε (x n ) ⇒ v ε . Now choose ε m 0 and apply the diagonal procedure to extract a subsequence n such that for each m ∈ N we have along {n } Corollary 2.10 in [13] v For later purposes we may write v m − x 0 ∈ B εm , where B r = {x ∈ D ; x ∞ ≤ r} for r > 0. Our final task consists in proving x n −→ σ x 0 . Let V be a σ-open neighborhood of x 0 . By the linearity there exists a σ-open neighborhood W of 0 such that W + W ⊂ V − x 0 . Since σ is coarser than the uniform topology, there exists δ > 0 such that B 2δ ⊂ W . Let m be such that ε m < δ. Then for n large enough we have  This is not so, if we admit atoms for µ, for (4.3) is then violated.

Remark 4.4.
In Introduction we suggested that the S topology is almost finer than any linear topology which is coarser than Skorokhod's J 1 topology. The delicate point is that condition (4.3) does not hold for J 1 . The corresponding typical example is given in Figure 2, with parameters t n → 0. To overcome this difficulty we shall introduce a variant of the J 1 topology, called mJ 1 (m -for modified), which slightly weakens the original topology and for which condition (4.3) is satisfied.
given by the formula Take the complete metric d Sk on D([−ε, T + ε]) (see [6])) and define d(x, y) = d Sk x, y .
Then (D, d) becomes a metric space and the corresponding topology will be called mJ 1 .    . Every linear topology on D, which is coarser than mJ 1 , is coarser than the S topology as well.
Remark 4.8. Were D, S a linear topological space, S would be the finest linear topology on D "below" mJ 1 .

Infinite time horizon
The problem consists in defining an analog of the S topology on the Skorokhod space D [0, +∞) of functions x : R + → R 1 , which are right-continuous at every t ≥ 0 and admit left limits at every t > 0. This cannot be achieved by invoking consistency, because the natural projections of D [0, T 2 ] , S onto D [0, T 1 ] , S , 0 < T 1 < T 2 , are not continuous, due to the special role of the end point T 1 ∈ (0, T 2 ).
A similar phenomenon was encountered long time ago for Skorokhod's J 1 topology (see [26] and [37] for the ways to overcome this difficulty). The case of the S topology can be handled in a somewhat simpler manner, mainly due to the characterization of * −→ S on D given in Theorem 3.3 and the fact that we are interested in convergence of sequences only and not in a particular form of a metric.  x 0 (t) dA 0 (t), The S topology on D [0, +∞) is the sequential topology generated by the L *convergence x n * −→ S x 0 . If x ∈ D [0, +∞) and T > 0, it will be convenient to denote by x T ∈ D [0, T ] the restriction of x to [0, T ]: x We have the following analog of Theorem 2.3.
We can find in every sequence {x n } of elements of K a subsequence {x n k } such that x n k * −→ S x 0 , as k → ∞, if, and only if, one of the following equivalent statements (i) and (ii) is satisfied.
Proof. The equivalence of (i) and (ii) is stated in Theorem 2.3, so it is enough to deal with (i) only.
Necessity. Suppose that for some T > 0 and along a sequence {x n } ⊂ K we have either lim n→∞ sup t∈[0,T ] |x n (t)| = +∞ or lim n→∞ N a,b x T n = +∞ for some a < b. Choose T > T . The sequence {x n } contains a subsequence x n k * −→ S x 0 and so We have arrived to a contradiction.
Sufficiency. Take any sequence T r +∞ and assume (i). By Theorem 2.3 we can find a sequence {x 1,n } such that x T1 1,n −→ S x 1,0 in D [0, T 1 ] . In {x 1,n } we can find a subsequence {x 2,n } such that x T2 2,n −→ S x 2,0 in D [0, T 2 ] . Repeating this process and then applying the diagonal procedure we can find a subsequence {x n } ⊂ K such that for every r ∈ N x Tr We claim that there exists exactly one x 0 ∈ D [0, +∞) such that for each r ∈ R x Tr 0 (t) = x r,0 (t), t ∈ [0, T r ).
Let q < r. It is enough to verify the consistency of x q,0 and x r,0 on [0, T q ). By Corollary 2.9 in [13] we can find a further subsequence {n } as well as countable subsets D q ⊂ [0, T q ) and D r ⊂ [0, T r ) such that It follows that if t belongs to the set [0, Because both x q,0 and x Tq r,0 are càdlàg, they are equal on [0, T q ). It remains to show that (5.1) holds for x n and x 0 . Let T > 0. Take T r ≥ T . By Notice that in view of the continuity of A 0 , the value of x r,0 at t = T r does not contribute to the value of the integral and therefore the limit integral can be written as hence the general case is implied by the one already proved. x r,0 (T r ) = x 0 (T r ), r ∈ R.
Let us repeat that construction for some T r +∞ and find a subsequence n such that Because (5.1) identifies the limit almost everywhere (as was shown in the proof of Theorem 3.3), we see that x r,0 (t) = x 0 (t), t ∈ [0, T r ), r ∈ N. Similarly as before, passing to a further subsequence {n } we have for some countable set D ⊂ R + x n (t) → x 0 (t), t ∈ D.
Take any T r ∈ (T r−1 , T r ] \ D. Then {x Tr n } is relatively S-compact in D [0, T r ] and x n (t) → x 0 (t) on a dense set containing T r . This implies x n −→ S x 0 in D [0, T r ] (for if not, we would be able to show that {x Tr n } is not relatively S-compact in D [0, T r ] , just as in the proof of Proposition 2.14 in [13]). 6 Appendix: Sequential topologies generated by L-convergences Following Fréchet, we say that X is a space of type L, if among all sequences of elements of X a class C(→) of "convergent" sequences is distinguished in such a way that: (i) To each convergent sequence (x n ) exactly one point x 0 , called "the limit", is attached (symbolically: x n −→ x 0 ) (ii) For every x ∈ X , the constant sequence (x, x, . . .) is convergent to x.
Using the L-convergence −→ one creates the family of closed sets. Definition 6.1. Say that F ⊂ X is τ (→)-closed if limits of −→-convergent sequences of elements of F remain in F , i.e. if x n ∈ F, n ∈ N and x n −→ x 0 , then x 0 ∈ F . The topology given by τ (→)-closed sets is called the sequential topology generated by the L-convergence −→ and will be denoted by τ (→).
Remark 6.2. It must be stressed that for a sequential topology to be defined only the extremely simple properties (ii) and (iii) of convergence −→ are required. On the other hand, the topology obtained this way has in general extremely poor separation properties.
It is only T1 space due to the fact that in view of (ii) above each one-point set {x} is τ (→)-closed. But it is enough for the topology τ (→) to define a new (in general) convergence, "−→ τ (→) " say, which, after Urysohn, is called the convergence "a posteriori", in order to distinguish from the original convergence (= convergence "a priori", i.e. "−→"). So (x n ) converges a posteriori to x 0 , if for every τ (→)-open set U containing x 0 eventually all elements of the sequence (x n ) belong to U .
Kantorovich et al [17, Theorem 2.42, p.51] and Kisyński [18] gave a familiar characterization of the convergence a posteriori in terms of the convergence a priori. Theorem 6.3 (KVPK recipe). {x n } converges to x 0 a posteriori if, and only if, each subsequence {x n k } contains a further subsequence {x n k l } convergent to x 0 a priori. Remark 6.4. The convergence a posteriori is generated by a topology. Suppose an L-convergence −→ satisfies additionally (iv) If every subsequence (x n k ) of (x n ) contains a further subsequence (x n k l ) −→convergent to x 0 , then the whole sequence (x n ) is −→-convergent to x 0 .
Then the convergence −→ is called an L * -convergence. It is an immediate consequence of Theorem 6.3 that if we start with an L * -convergence then the convergences a posteriori and a priori coincide.
Remark 6.5. It follows that given an L-convergence "−→" we can weaken it to an L * -convergence " * −→" which is already the usual convergence of sequences in the topological space (X , τ (−→)) ≡ (X , τ ( * →)). At least two examples of such a procedure are commonly known. Example 6.6. If "−→" denotes the convergence "almost surely" of real random variables defined on a probability space (Ω, F, P), then " * −→" is the convergence "in probability". Example 6.7. Let X = R 1 and take a sequence ε n 0. Say that x n −→ x 0 , if for each n ∈ N, |x n − x 0 | < ε n , i.e. x n converges to x 0 at the given rate {ε n }. Then " * −→" means the usual convergence of real numbers. Remark 6.8. It is worth noting that a set J ⊂ X is relatively −→-compact (i.e. in each sequence {x n } ⊂ J one can find a subsequence {n } such that x n −→ x 0 , for some x 0 ∈ X ) iff it is relatively * −→-compact. Remark 6.9. Let us notice that if (X , τ ) is a Hausdorff topological space, then τ ⊂ τ s ≡ τ ( −→ τ ) and in general this inclusion may be strict (like in the case of the weak topology on an infinite dimensional Hilbert space).