A class W of probability measures defined on ( R + 2 , B ( R + 2 ) ) is introduced in [4]. It consists of all probability measures satisfying 1)

∫ R + 2 ( x + y ) μ ( d x , d y ) < ∞,

The procedure of constructing a triple V, W and Z for a measure μ ∈ W is available from the proof of Proposition 3.6 [4]. For measures μ ∈ W with finite support, calculations are explicit. For example, let μ = 1 2 δ ( 1 , 3 / 2 ) + 1 2 δ ( 3 / 2 , 1 ) . Then μ ∈ W. Put Ω : = { 0 , 1 } × [ 0 ; 1 ], F : = B ( Ω ), P ( { 0 } × B ) = 1 2 Λ ( B ) , P ( { 1 } × B ) = 1 2 Λ ( B ) , where B ∈ B ( [ 0 ; 1 ] ) and Λ is the Lebesgue measure on [ 0 ; 1 ]. Then one can take V ( 0 , u ) : = 1 2 − u , V ( 1 , u ) : = 0 , W ( 0 , u ) : = u 4 − 2 u , W ( 1 , u ) : = 1 / 2 , Z ( 0 , u ) : = 1 − u 4 − 2 u , Z ( 1 , u ) : = 1 .

Our aim here is to show that measures with finite support and lying in W are dense in W in some natural topology of the space of Borel measures μ ( d x , d y ) on ( R 2 , B ( R 2 ) ) with integrable marginal distributions.

We will make a few remarks that are intended to demonstrate that our task is nontrivial. If E [ V 1 { W ≤ t } ] = E [ W ∧ t ] for all t ≥ 0, then it is easy to see that the distribution of W cannot be discrete. Note that in the example above W has a continuous component, though the distribution μ has a finite support. This means that the subset of W consisting of measures with equality in 3) does not contain measures with finite support. In particular, if V = 1, then W has the exponential distribution with mean 1. Example 2.1 in [4] says that if we consider a two-point mass ν that is obtained from the exponential distribution by averaging over the sets [ 0 ; a ] and ( a ; + ∞ ), where a > 00$]]>, then the measure δ { 1 } ⊗ ν does not belong to W.

The problem considered here is an important technical tool in studying some stochastic orders on the plane. In particular, let μ 0 ( d x , d y ) ≼ μ 1 ( d x , d y ) if there are stochastic processes X = ( X t ) t ∈ [ 0 ; 1 ] and Y = ( Y t ) t ∈ [ 0 ; 1 ] such that X is adapted and Y is predictable on some stochastic basis, X t and Y t are integrable for all t, X t − X 0 is an increasing process with the compensator Y t − Y 0 , and Law ( X 0 , Y 0 ) = μ 0 and Law ( X 1 , Y 1 ) = μ 1 . In these terms μ ∈ W is equivalent to δ ( 0 , 0 ) ≼ μ. Our main result allows us to find necessary and sufficient conditions for μ 0 ≼ μ 1 expressed in appropriate terms [1].

Let us also mention a connection between our problem and the Skorokhod embedding problem. Namely, let B be a standard Brownian motion and T a finite stopping time on some stochastic basis. Put B ‾ T : = sup t ≤ T B t . Then, if E | B T | < ∞ and E [ B ‾ T ] < ∞, Law ( B ‾ T − B T , B ‾ T ) ∈ W. However, not every distribution μ from W can be represented in this way. The necessary and sufficient condition on μ is that R + ∋ t ↦ ∫ R + 2 [ ( y ∧ t ) − x 1 { y ≤ t } ] μ ( d x , d y ) be an increasing function and μ ( ( 0 ; ∞ ) × { 0 } ) = 0, see [8] and [6]. A discretisation of the Skorokhod embedding problem is considered, in particular, in [2].

To state the problem we need a proper topological setting. Consider a gauge function ψ ( x , y ) : = 1 + | x | + | y |, x , y ∈ R, and define the class C ψ ( R 2 ) of continuous test functions f : R 2 → R such that ∀ f ∈ C ψ ( R 2 ) ∃ c ∈ R ∀ x , y ∈ R | f ( x , y ) | ≤ c · ψ ( x , y ) . Denote also by M 1 ψ ( R 2 ) the set of all Borel measures on R 2 such that ∫ R 2 ψ ( x , y ) μ ( d x , d y ) < ∞ .

The coarsest topology on M 1 ψ ( R 2 ) for which all mappings M 1 ψ ( R 2 ) ∋ μ ↦ ∫ R 2 f ( x , y ) μ ( d x , d y ) , f ∈ C ψ ( R 2 ) , are continuous, is called the ψ-weak topology on M 1 ψ ( R 2 ).

We refer to [3, §A.6] for the definition and further properties of ψ-weak topologies with arbitrary gauge functions ψ. Here we only mention that in our case the ψ-weak topology is metrizable and separable. Moreover, M 1 ψ ( R 2 ) is a Polish space with this topology. A sequence ( μ n ) n = 1 ∞ ⊆ M 1 ψ ( R 2 ) converges to a measure μ ∈ M 1 ψ ( R 2 ) in the ψ-weak topology if and only if ( μ n ) n = 1 ∞ converges to μ in the usual weak topology and the uniform integrability condition sup n ∈ N ∫ { | x | + | y | > c } ( | x | + | y | ) μ n ( d x , d y ) → 0 as c → ∞ c\}}}(|x|+|y|)\hspace{0.1667em}{\mu _{n}}(dx,\hspace{0.1667em}dy)\to 0\hspace{10pt}\text{as}\hspace{5pt}c\to \infty \]]]> is satisfied.

The choice of the ψ-weak topology with the prescribed function ψ is natural in this problem. On the one hand, the space M 1 ψ ( R 2 ) contains the set W. On the other hand, it is this topology that insures the closedness of the set W in M 1 ψ ( R 2 ), while the usual weak topology does not provide the closedness.

The proof of our main result is based on Theorem 5 in [5]. For the reader’s convenience, we present this theorem here with some refinement in the statement. We do not explain here how a single jump filtration is defined because the only fact concerning these filtrations that we use is that such a filtration exists.

The only difference between this theorem and Theorem 5 in [5] refers to the case where γ is bounded by a constant t G < ∞ with probability one and P { γ = t G } > 00$]]>. In this case, in Theorem 5 in [5] F is defined by left-continuity at t G , so the difference between the definitions is ∫ { t G } G ‾ ( s − ) − 1 K ( s ) d G ( s ) = K ( t G ) . Now it is easy to see that the formulae for the compensators from both theorems coincide. Thus, our definition of F at t G allows us to combine two terms in the formula for the compensator in Theorem 5 in [5] and to obtain a unified expression.

Let us consider two subsets of the set W, namely, the subset of simple measures W simp and the subset of discrete measures W disc . We say that μ ∈ W simp (correspondingly, μ ∈ W disc ) if μ ∈ W and μ ( d x , d y ) = ∑ j ∈ J p j · δ x j a j ( d x , d y ) , where J is a finite set (correspondingly, J is at most countable), p j ≥ 0, ∑ j ∈ J p j = 1, and δ x j a j ( d x , d y ) is the Dirac measure at point x j a j ∈ R 2 . It is clear that W simp ⊆ W disc ⊆ W.

The following theorem is the main result of this paper.

The proof will be performed in two steps:

Step 1: Given μ ∈ W, construct a sequence ( μ n ) n = 1 ∞ ⊆ W disc that converges to μ in the ψ-weak topology.

Step 2: Using the sequence ( μ n ) n = 1 ∞ from Step 1, construct a sequence ( ν n ) n = 1 ∞ ⊆ W simp converging to μ in the same topology. Notice that most of the arguments of the proof are the same as at Step 1. Proof.

Fix a measure μ ∈ W. Due to [4, Proposition 3.6, Remark 3.4, and Proposition 3.4], one can find a stochastic basis B : = ( Ω , F , P , ( G t ) t ∈ R + ) and nonnegative integrable random variables V, W, Z such that (1) Law ( V + Z , W + Z ) = μ , and the process V 1 { W ≤ t }, t ∈ R + , is an integrable increasing process with the compensator W ∧ t, t ∈ R + . Therefore, E [ V 1 { W ≤ t } ] = E [ W ∧ t ] for all t ∈ R + , i.e., condition 3) from the definition of W holds as equality. In particular, Law ( V , W ) ∈ W.

Step 1. It is enough to construct three sequences of nonnegative random variables ( V ˆ ( n ) ) n = 1 ∞ , ( W ˆ ( n ) ) n = 1 ∞ , and ( Z ˆ ( n ) ) n = 1 ∞ on the probability space ( Ω , F , P ) which satisfy the following conditions: (i)

Law ( V ˆ ( n ) , W ˆ ( n ) ) ∈ W disc for all n ∈ N;

In what follows we will use a result from [5]. Let us introduce some notation from that paper. Denote •

G ( t ) : = P { W ≤ t }, t ∈ R;

It is easy to see that P { W ∉ T } = 0 and T = [ 0 ; t G ) in Cases A1 and A2, and T = [ 0 ; t G ] in Case B.

Let us define auxiliary random variables V ( n ) : = ∑ i = 1 ∞ i Δ 2 n 1 { ( i − 1 ) Δ 2 n < V ≤ i Δ 2 n } , W ( n ) : = ∑ j = 1 ∞ j Δ 2 n 1 { ( j − 1 ) Δ 2 n < W ≤ j Δ 2 n } , Z ( n ) : = ∑ k = 1 ∞ k Δ 2 n 1 { ( k − 1 ) Δ 2 n < Z ≤ k Δ 2 n } , where Δ : = 1 if t G = ∞, and Δ : = t G if t G < ∞.

Similarly to how we defined G ( t ), G ‾ ( t ), t G and T for W, let us define G ( n ) ( t ), G ‾ ( n ) ( t ), t G ( n ) , and T ( n ) for W ( n ) : •

G ( n ) ( t ) : = P { W ( n ) ≤ t }, t ∈ R;

It is easy to see that, for any n ∈ N, t G ( n ) = t G and P { W ( n ) ∉ T ( n ) } = 0; T ( n ) = [ 0 ; t G ] in Cases A1 and B; and T ( n ) = [ 0 ; t G ) in Case A2.

Finally, let us introduce •

K ( n ) ( t ) : = E V ( n ) | W ( n ) = t , where t ∈ T ( n ) ;

Fix n ∈ N. Introduce the single jump filtration generated by W ( n ) and F, see [5]. In accordance with Theorem 1 the process V ( n ) 1 { W ( n ) ≤ t }, t ∈ R + , has the compensator F ( n ) ( W ( n ) ∧ t ), t ∈ R + , with respect to this filtration.

Denote by V ˆ ( n ) and W ˆ ( n ) the values of this increasing process and its compensator at time ∞: V ˆ ( n ) : = lim t → ∞ V ( n ) 1 { W ( n ) ≤ t } = V ( n ) , W ˆ ( n ) : = lim t → ∞ F ( n ) ( W ( n ) ∧ t ) = F ( n ) ( W ( n ) ) . Put also Z ˆ ( n ) : = Z ( n ) . Obviously, V ˆ ( n ) , W ˆ ( n ) , and Z ˆ ( n ) are discrete random variables. By the construction, Law ( V ˆ ( n ) , W ˆ ( n ) ) ∈ W. Thus, taking into account Lemma 1 in Section 3, Law ( V ˆ ( n ) + Z ˆ ( n ) , W ˆ ( n ) + Z ˆ ( n ) ) ∈ W. It follows that our construction satisfies (i) and (ii). Conditions (iii) and (iv) are clear from the definitions.

The proof of condition (v) is the main point of Step 1. We defer the proof to the next section, see Lemma 4.

Assuming (v), we complete the proof of Step 1. We have E [ W ˆ ( n ) ] = E [ V ˆ ( n ) ] → E [ V ] = E [ W ] as n → ∞ , and (vi) follows because W ˆ ( n ) are nonnegative and integrable.

Step 2. In contrast to Step 1 we define auxiliary random variables V [ n ] , W [ n ] and Z [ n ] taking finite number of values instead of discrete random variables V ( n ) , W ( n ) and Z ( n ) : V [ n ] : = ∑ i = 1 r ( n ) i Δ 2 n 1 { ( i − 1 ) Δ 2 n < V ≤ i Δ 2 n } , W [ n ] : = ∑ j = 1 r ( n ) j Δ 2 n 1 { ( j − 1 ) Δ 2 n < W ≤ j Δ 2 n } + ( r ( n ) + 1 ) Δ 2 n 1 { W > r ( n ) Δ 2 n } , Z [ n ] : = ∑ k = 1 r ( n ) k Δ 2 n 1 { ( k − 1 ) Δ 2 n < Z ≤ k Δ 2 n } , \frac{r(n)\Delta }{{2^{n}}}\Big\}\text{,}\\ {} {Z^{[n]}}& :={\sum \limits_{k=1}^{r(n)}}\frac{k\Delta }{{2^{n}}}\mathbb{1}\Big\{\frac{(k-1)\Delta }{{2^{n}}} where Δ : = 1 if t G = ∞, and Δ : = t G if t G < ∞, and r : N → N is a sequence such that (3) ∀ n ∈ N r ( n ) ≥ 2 n and r ( n ) 2 n ↑ ∞ as n → ∞ . The choice of r ( n ) will be specified in Lemma 5.

Similarly to Step 1, we define •

G [ n ] ( t ) : = P { W [ n ] ≤ t }, t ∈ R;

Our aim is to prove the statements (i)–(vi) from Step 1 for V ˆ [ n ] , W ˆ [ n ] and Z ˆ [ n ] instead of V ˆ ( n ) , W ˆ ( n ) and Z ˆ ( n ) . The proof of (i)–(iv), (vi) is similar to Step 1. We only mention that V ˆ [ n ] , W ˆ [ n ] and Z ˆ [ n ] take values in finite sets and our assumptions on r ( n ) imply (iii). The only significant difference between Steps 1 and 2 is in the proof of (v). In the course of the proof we will replace Lemmas 3 and 4 by Lemmas 5 and 6 respectively.  □