Abstract. Let µ

FOR UNIFORM DISTANCE

PATRIZIA BERTI, LUCA PRATELLI, AND PIETRO RIGO

Abstract. Let µ

be a probability measure on the Borel σ-field on D[0, 1]

with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X

such that X

∼ µ

for all n ≥ 0 and kX

− X

k → 0 in probability, where k·k is the sup-norm. Such conditions do not require µ

separable under k·k. Applications to exchangeable empirical processes and to pure jump processes are given as well.

1. Introduction Let D be the set of real cadlag functions on [0, 1] and

kxk = sup

t

|x(t)|, u(x, y) = kx − yk, x, y ∈ D.

Also, let d be Skorohod distance and B d , B u the Borel σ-fields on D with respect to (w.r.t.) d and u, respectively.

In real problems, one usually starts with a sequence (µ n : n ≥ 0) of probabilities on B d . If µ n → µ 0 weakly (under d), Skorohod representation theorem yields d(X n , X 0 ) −→ 0 for some D-valued random variables X ^a.s. n such that X n ∼ µ n for all n ≥ 0. However, X _n can fail to approximate X ₀ uniformly. A trivial example is µ _n = δ _x

, where (x _n ) ⊂ D is any sequence such that x _n → x 0 according to d but not according to u.

Lack of uniform convergence is sometimes a trouble. Thus, given a sequence (µ _n : n ≥ 0) of laws on B _d , it is useful to have conditions for:

On some probability space (Ω, A, P ), there are random variables X _n : Ω → D such that X _n ∼ µ _n for all n ≥ 0 and kX _n − X ₀ k −→ 0. ^P (1)

Convergence in probability cannot be strengthened into a.s. convergence in condi- tion (1). In fact, it may be that (1) holds, and yet there are not D-valued random variables Y n such that Y n ∼ µ n for all n and kY n − Y 0 k −→ 0; see Example 7. ^a.s.

This paper is concerned with (1). The main result is Theorem 4, which states that (1) holds if and only if

(2) lim

n sup

f ∈L

|µ _n (f ) − µ ₀ (f )| = 0,

2000 Mathematics Subject Classification. 60B10, 60A05, 60A10.

Key words and phrases. Cadlag function – Exchangeable empirical process – Separable prob- ability measure – Skorohod representation theorem– Uniform distance – Weak convergence of probability measures.

1

where L is the set of functions f : D → R satisfying

σ(f ) ⊂ B d , −1 ≤ f ≤ 1, |f (x) − f (y)| ≤ kx − yk for all x, y ∈ D.

Theorem 4 can be commented as follows. Say that a probability µ, defined on B d or B u , is u-separable in case µ(A) = 1 for some u-separable A ∈ B d . Suppose µ 0 is u-separable and define µ ^∗ ₀ (H) = µ 0 (A ∩ H) for H ∈ B u , where A ∈ B d is u-separable and µ 0 (A) = 1. Since µ n is defined only on B d for n ≥ 1, we adopt Hoffmann-Jørgensen’s definition of convergence in distribution for non measurable random elements; see e.g. [7] and [9]. Let I 0 be the identity map on (D, B u , µ ^∗ ₀ ) and I n the identity map on (D, B d , µ n ), n ≥ 1. Further, let D be regarded as a metric space under u. Then, since µ ^∗ ₀ is u-separable, one obtains:

(i) Condition (1) holds (with kX n −X 0 k −→ 0) provided I ^a.s. n → I 0 in distribution;

(ii) I _n → I ₀ in distribution if and only if lim _n sup _{f ∈L} |µ _n (f ) − µ ₀ (f )| = 0.

Both (i) and (ii) are known facts; see Theorems 1.7.2, 1.10.3 and 1.12.1 of [9].

The spirit of Theorem 4, thus, is that one can dispense with u-separability of µ 0 to get (1). This can look surprising, as separability of the limit law is crucial in Skorohod representation theorem; see [5]. However, X n ∼ µ n is asked only on B d

and not on B u . Indeed, X n can even fail to be measurable w.r.t. B u .

Non u-separable laws on B d are quite usual. A cadlag process Z, with jumps at random time points, has typically a non u-separable distribution on B d . One example is Z(t) = B _{M (t)} , where B is a standard Brownian bridge, M an indepen- dent random distribution function and the jump-points of M have a non discrete distribution. Such a Z is the limit in distribution, under d, of certain exchangeable empirical processes; see [1] and [3].

In applications, unless µ ₀ is u-separable, checking condition (2) is usually diffi- cult. In this sense, Theorem 4 can be viewed as a ”negative” result, as it states that condition (1) is quite hard to reach. This is partly true. However, there are also meaningful situations where (2) can be proved with a reasonable effort. Two examples are exchangeable empirical processes, which motivated Theorem 4, and a certain class of jump processes. Both are discussed in Section 4.

Our proof of Theorem 4 is admittedly long and it is confined in a final appendix.

Some preliminary results, of possible independent interest, are needed. We mention Proposition 2 and Lemma 13 in particular.

A last remark is that Theorem 4 is still valid if D is replaced by D [0, 1], X
, the space of cadlag functions from [0, 1] into a separable Banach space X .

2. A preliminary result

Let (Ω, A, P ) be a probability space. The outer and inner measures are P ^∗ (H) = inf{P (A) : H ⊂ A ∈ A}, P _∗ (H) = 1 − P ^∗ (H ^c ), H ⊂ Ω.

Given a metric space (S, ρ) and maps X _n : Ω → S, n ≥ 0, say that X _n converges to X 0 in (outer) probability, written X n

−→ X P 0 , in case lim n P ^∗ ρ(X n , X 0 ) > 
= 0 for all  > 0.

In the sequel, d _{T V} denotes total variation distance between two probabilities

defined on the same σ-field.

Proposition 1. Let (F, F ) be a measurable space and µ n a probability on (F, F ), n ≥ 0. Then, on some probability space (Ω, A, P ), there are measurable maps X n : (Ω, A) → (F, F ) such that

P _∗ (X _n 6= X ₀ ) = P ^∗ (X _n 6= X ₀ ) = d _{T V} (µ _n , µ ₀ ) and X _n ∼ µ _n for all n ≥ 0.

Proposition 1 is well known, even if in a slightly different form; see Theorem 2.1 of [8]. A proof of the present version is in Section 3 of [5].

Next proposition is fundamental for proving our main result. Among other things, it can be viewed as an improvement of Proposition 1.

Proposition 2. Let λ _n be a probability on (F × G, F ⊗ G), n ≥ 0, where (F, F ) is a measurable space and (G, G) a Polish space equipped with its Borel σ-field. The following conditions are equivalent:

(a) There are a probability space (Ω, A, P ) and measurable maps (Y n , Z n ) : (Ω, A) −→ (F × G, F ⊗ G) such that

(Y n , Z n ) ∼ λ n for all n ≥ 0, P ^∗ (Y n 6= Y 0 ) −→ 0, Z n

−→ Z P 0 ; (b) For each bounded Lipschitz function f : G → R,

lim n sup

A∈F

   Z

I _A (y) f (z) λ _n (dy, dz) − Z

I _A (y) f (z) λ ₀ (dy, dz)    = 0.

To prove Proposition 2, we first recall a result of Blackwell and Dubins [6].

Theorem 3. Let G be a Polish space, M the collection of Borel probabilities on G, and m the Lebesgue measure on (0, 1). There is a Borel measurable map

Φ : M × (0, 1) −→ G such that, for every ν ∈ M,

(i) Φ(ν, ·) ∼ ν under m;

(ii) There is a Borel set A _ν ⊂ (0, 1) such that m(A ν ) = 1 and

Φ(ν n , t) −→ Φ(ν, t) whenever t ∈ A ν , ν n ∈ M and ν n → ν weakly.

We also need to recall disintegrations. Let λ be a probability on (F × G, F ⊗ G), where (F, F ) and (G, G) are arbitrary measurable spaces. In this paper, λ is said to be disintegrable if there is a collection α = {α(y) : y ∈ F } such that:

− α(y) is a probability on G for y ∈ F ;

− y 7→ α(y)(C) is F -measurable for C ∈ G;

− λ(A × C) = R

A α(y)(C) µ(dy) for A ∈ F and C ∈ G, where µ(·) = λ(· × G).

Such an α is called a disintegration for λ. For λ to admit a disintegration, it suffices that G is a Borel subset of a Polish space and G the Borel σ-field on G.

Proof of Proposition 2. ”(a) ⇒ (b)”. Under (a), for each A ∈ F and bounded Lipschitz f : G → R, one obtains

   Z

I A (y) f (z) λ n (dy, dz) − Z

I A (y) f (z) λ 0 (dy, dz)    =

 E P I A (Y n ) f (Z n ) − I A (Y 0 ) f (Z 0 )   

≤ E P

 f (Z n ) (I A (Y n ) − I A (Y 0 ))    + E P

 I A (Y 0 ) (f (Z n ) − f (Z 0 ))   

≤ sup|f | P ^∗ (Y n 6= Y 0 ) + E P

 f (Z n ) − f (Z 0 )

 −→ 0.

”(b) ⇒ (a)”. Let µ n (A) = λ n (A × G), A ∈ F . By (b), d T V (µ n , µ 0 ) → 0.

Hence, by Proposition 1, on a probability space (Θ, E , Q) there are measurable maps h n : (Θ, E ) → (F, F ) satisfying h n ∼ µ n for all n and Q ^∗ (h n 6= h 0 ) → 0. Let

Ω = Θ × (0, 1), A = E ⊗ B _(0,1) , P = Q × m, where B _(0,1) is the Borel σ-field on (0, 1) and m the Lebesgue measure.

Since G is Polish, each λ _n admits a disintegration α _n = {α _n (y) : y ∈ F }. By Theorem 3, there is a map Φ : M × (0, 1) −→ G satisfying conditions (i)-(ii). Let

Y n (θ, t) = h n (θ) and Z n (θ, t) = Φα n (h n (θ)), t , (θ, t) ∈ Θ × (0, 1).

For fixed θ, condition (i) yields Z n (θ, ·) = Φα n (h n (θ)), · ∼ α n (h n (θ)) under m.

Since α n is a disintegration for λ n , for all A ∈ F and C ∈ G one has P (Y n ∈ A, Z n ∈ C) =

Z

Θ

I A (h n (θ)) mt : Z n (θ, t) ∈ C Q(dθ)

= Z

{h

∈A}

α n (h n (θ))(C) Q(dθ) = Z

A

α n (y)(C) µ n (dy) = λ n (A × C).

Also, P ^∗ (Y _n 6= Y ₀ ) = Q ^∗ (h _n 6= h ₀ ) −→ 0 by Lemma 1.2.5 of [9].

Finally, we prove Z n

−→ Z P 0 . Write α n (y)(f ) = R f (z) α n (y)(dz) for all y ∈ F and f ∈ L _G , where L _G is the set of Lipschitz functions f : G → [−1, 1]. Since Q ^∗ (h _n 6= h 0 ) → 0, there are A _n ∈ F such that Q(A ^c _n ) → 0 and h _n = h ₀ on A _n . Given f ∈ L _G ,

E _Q  

 α _n (h _n )(f ) − α ₀ (h ₀ )(f )  

 − 2 Q(A ^c _n ) ≤ E _Q I A

 α _n (h ₀ )(f ) − α ₀ (h ₀ )(f )   

≤ E Q

 α _n (h ₀ )(f ) − α ₀ (h ₀ )(f )    =

Z  

 α _n (y)(f ) − α ₀ (y)(f )    µ ₀ (dy).

Using condition (b), it is not hard to see that R |α n (y)(f ) − α ₀ (y)(f )| µ ₀ (dy) −→ 0.

Therefore, α _n (h _n )(f ) −→ α ^Q ₀ (h ₀ )(f ) for each f ∈ L _G , and this is equivalent to each subsequence (n ⁰ ) contains a further subsequence (n ⁰⁰ )

such that α n

(h n

(θ)) −→ α 0 (h 0 (θ)) weakly for Q-almost all θ;

see Remark 2.3 and Corollary 2.4 of [2]. Thus, by property (ii) of Φ, each subse- quence (n ⁰ ) contains a further subsequence (n ⁰⁰ ) such that Z n

−→ Z a.s. 0 . That is, Z _n −→ Z ^P 0 and this concludes the proof.

 3. Existence of cadlag processes, with given distributions on the

Skorohod Borel σ-field, converging uniformly in probability As in Section 1, B _d and B _u are the Borel σ-fields on D w.r.t. d and u. Also, L is the class of functions f : D → [−1, 1] which are measurable w.r.t. B _d and Lipschitz w.r.t. u with Lipschitz constant 1. We recall that, for x, y ∈ D, the Skorohod distance d(x, y) is the infimum of those  > 0 such that

kx − y ◦ γk ≤  and sup

s6=t

 log γ(s) − γ(t) s − t

   ≤ 

for some strictly increasing homeomorphism γ : [0, 1] → [0, 1]. The metric space

(D, d) is separable and complete.

We write µ(f ) = R f dµ whenever µ is a probability on a σ-field and f a real bounded function, measurable w.r.t. such a σ-field.

Motivations for the next result have been given in Section 1.

Theorem 4. Let µ _n be a probability measure on B _d , n ≥ 0. Then, conditions (1) and (2) are equivalent, that is,

lim n sup

f ∈L

|µ _n (f ) − µ ₀ (f )| = 0

if and only if there are a probability space (Ω, A, P ) and measurable maps X _n : (Ω, A) → (D, B _d ) such that X _n ∼ µ _n for each n ≥ 0 and kX _n − X ₀ k −→ 0. ^P

The proof of Theorem 4 is given in the Appendix. Here, we state a corollary and an open problem and we make two examples.

In applications, the µ n are often probability distributions of random variables, all defined on some probability space (Ω 0 , A 0 , P 0 ). In the spirit of [4], a (minor) question is whether condition (1) holds with the X n defined on (Ω 0 , A 0 , P 0 ) as well.

Corollary 5. Let (Ω 0 , A 0 , P 0 ) be a probability space and Z n : (Ω 0 , A 0 ) → (D, B d ) a measurable map, n ≥ 1. Suppose lim n sup _{f ∈L} |E P

f (Z n ) − µ 0 (f )| = 0 for some probability measure µ 0 on B d . If P 0 is nonatomic, there are measurable maps X n : (Ω 0 , A 0 ) → (D, B d ), n ≥ 0, such that

X 0 ∼ µ 0 , X n ∼ Z n for each n ≥ 1, kX n − X 0 k −→ 0. ^P

Also, P 0 is nonatomic if µ 0 {x} = 0 for all x ∈ D, or if P 0 (Z n = x) = 0 for some n ≥ 1 and all x ∈ D.

Proof. Since (D, d) is separable, P 0 is nonatomic if P 0 (Z n = x) = 0 for some n ≥ 1 and all x ∈ D. By Corollary 5.4 of [4], (Ω 0 , A 0 , P 0 ) supports a D-valued random variable Z 0 with Z 0 ∼ µ 0 . Hence, P 0 is nonatomic even if µ 0 {x} = 0 for all x ∈ D.

Next, by Theorem 4, on a probability space (Ω, A, P ) there are D-valued random variables Y n such that Y 0 ∼ µ 0 , Y n ∼ Z n for n ≥ 1 and kY n − Y 0 k −→ 0. Let ^P (D ^∞ , B ^∞ _d ) be the countable product of (D, B d ) and

ν(A) = P (Y 0 , Y 1 , . . .) ∈ A
, A ∈ B _d ^∞ .

Then, ν is a Borel probability on a Polish space. Thus, if P 0 is nonatomic, (Ω 0 , A 0 , P 0 ) supports a D ^∞ -valued random variable X = (X 0 , X 1 , . . .) with X ∼ ν;

see e.g. Theorem 3.1 of [4]. Since (X 0 , X 1 , . . .) ∼ (Y 0 , Y 1 , . . .), this concludes the

proof. 

Let (S, ρ) be a metric space such that (x, y) 7→ ρ(x, y) is measurable w.r.t. E ⊗ E , where E is the ball σ-field on S. This is actually true in case (S, ρ) = (D, u) and it is very useful to prove Theorem 4. Thus, a question is whether (D, u) can be replaced by (S, ρ) in Theorem 4. Precisely, let (µ n : n ≥ 0) be a sequence of laws on E and L S the class of functions f : S → [−1, 1] such that σ(f ) ⊂ E and

|f (x) − f (y)| ≤ ρ(x, y) for all x, y ∈ S. Then,

Conjecture: lim n sup _{f ∈L}

|µ n (f ) − µ 0 (f )| = 0 if and only if ρ(X n , X 0 ) −→ 0 in probability for some S-valued random variables X n such that X n ∼ µ n for all n.

We finally give two examples. The first shows that condition (2) cannot be

weakened into µ _n (f ) → µ ₀ (f ) for each fixed f ∈ L.

Example 6. For each n ≥ 0, let h n : (0, 1) → [0, ∞) be a Borel function such that R 1

0 h n (t) dt = 1. Suppose that h n → h 0 in σ(L 1 , L _∞ ) but not in L 1 under Lebesgue measure m on (0, 1), that is,

lim sup _n R 1

0 |h _n (t) − h ₀ (t)| dt > 0, lim n

R 1

0 h n (t) g(t) dt = R 1

0 h 0 (t) g(t) dt for all bounded Borel functions g.

(3)

Take a sequence (T n : n ≥ 0) of (0, 1)-valued random variables, on a probability space (Θ, E , Q), such that each T n has density h n w.r.t. m. Define

Z _n = I _[T

_,1] and µ _n (A) = Q(Z _n ∈ A) for A ∈ B _d .

Then Z _n = φ(T _n ), with φ : (0, 1) → D given by φ(t) = I _[t,1] , t ∈ (0, 1). Hence, for fixed f ∈ L, one obtains

µ n (f ) = E Q f ◦ φ(T n ) = Z 1

0

h n (t) f ◦ φ(t) dt −→

Z 1 0

h 0 (t) f ◦ φ(t) dt = µ 0 (f ).

Suppose now that X _n ∼ µ _n for all n ≥ 0, where the X _n are D-valued random variables on some probability space (Ω, A, P ). Since

P ω : X n (ω)(t) ∈ {0, 1} for all t = Qθ : Z n (θ)(t) ∈ {0, 1} for all t = 1, it follows that

P kX n − X 0 k > 1

2
= P (X n 6= X 0 ) ≥ d T V (µ n , µ 0 ) = 1 2

Z 1 0

|h n (t) − h 0 (t)| dt.

Therefore, X n fails to converge to X 0 in probability.

A slight change in Example 6 shows that convergence in probability cannot be strengthened into a.s. convergence in condition (1). Precisely, it may be that (1) holds, and yet there are not D-valued random variables Y n satisfying Y n ∼ µ n for all n and kY _n − Y 0 k −→ 0. ^a.s.

Example 7. In the notation of Example 6, instead of (3) assume

lim n

Z 1 0

|h n (t) − h 0 (t)| dt = 0 and m lim inf

n h n < h 0
> 0 where m is Lebesgue measure on (0, 1). Since

d _{T V} (µ _n , µ ₀ ) = 1 2

Z 1 0

|h n (t) − h ₀ (t)| dt −→ 0,

condition (1) trivially holds by Proposition 1. Suppose now that Y n ∼ µ n for all n ≥ 0, where the Y n are D-valued random variables on a probability space (Ω, A, P ).

As m lim inf _n h _n < h ₀
> 0, Theorem 3.1 of [8] yields P (Y n = Y ₀ ultimately) < 1.

On the other hand, since P Y n (t) ∈ {0, 1} for all t
= 1, one obtains

P kY _n − Y ₀ k −→ 0
= P (Y n = Y ₀ ultimately) < 1.

4. Applications

Condition (2) is not always hard to be checked, even if µ 0 is not u-separable.

We illustrate this fact by two examples. To this end, we first note that conditions (1)-(2) are preserved under certain mixtures.

Corollary 8. Let G be the set of distribution functions on [0, 1] and G the σ-field on G generated by the maps g 7→ g(t), 0 ≤ t ≤ 1. Let π be a probability on G and µ n and λ n probabilities on B d . Then, condition (1) holds provided

sup

f ∈L

|λ n (f ) − λ ₀ (f )| −→ 0 and

µ n (A) = Z

λ n x : x ◦ g ∈ A π(dg) for all n ≥ 0 and A ∈ B d .

Proof. By Theorem 4, there are a probability space (Θ, E , Q) and measurable maps Z _n : (Θ, E ) −→ (D, B _d ) such that Z _n ∼ λ _n for all n and kZ _n − Z ₀ k −→ 0. Define ^Q Ω = Θ × G, A = E ⊗ G, P = Q × π, and X _n (θ, g) = Z _n (θ) ◦ g for all (θ, g) ∈ Θ × G.

It is routine to check that X n ∼ µ n for all n and kX n − X 0 k −→ 0. ^P  Example 9. (Exchangeable empirical processes). Let (ξ _n : n ≥ 1) be a sequence of [0, 1]-valued random variables on the probability space (Ω ₀ , A ₀ , P ₀ ).

Suppose (ξ _n ) exchangeable and define

F (t) = E P

I _{ξ

_≤t} | τ

where τ is the tail σ-field of (ξ _n ). Take F to be regular, i.e., each F -path is a distribution function. Then, the n-th empirical process can be defined as

Z n (t) = P n

i=1 I {ξ

≤t} − F (t)

√ n , 0 ≤ t ≤ 1, n ≥ 1.

Since Z _n : (Ω ₀ , A ₀ ) → (D, B _d ) is measurable, one can define µ _n (·) = P ₀ (Z _n ∈ ·).

Also, let µ ₀ be the probability distribution of Z 0 (t) = B M (t)

where B is a standard Brownian bridge on [0, 1] and M an independent copy of F (with B and M defined on some probability space). Then, µ _n → µ ₀ weakly (under d) but µ 0 can fail to admit any extension to B u ; see [3] and Example 11 of [1]. Thus, Z n can fail to converge in distribution, under u, according to Hoffmann-Jørgensen’s definition. However, Corollaries 5 and 8 grant that:

On (Ω 0 , A 0 , P 0 ), there are measurable maps X n : (Ω 0 , A 0 ) → (D, B d ) such that X _n ∼ Z n for each n ≥ 0 and kX _n − X 0 k −→ 0. ^P

Define in fact B n (t) = n ^−1/2 P n

i=1 I {u

≤t} − t , where u 1 , u 2 , . . . are i.i.d. ran- dom variables (on some probability space) with uniform distribution on [0, 1]. Then, B n → B in distribution, under u, according to Hoffmann-Jørgensen’s definition. Let λ n and λ 0 be the probability distributions of B n and B, respectively. Since λ 0 is u-separable, sup _{f ∈L} |λ n (f ) − λ 0 (f )| −→ 0 (see Section 1). Thus, the first condition of Corollary 8 holds. The second condition follows from de Finetti’s representation theorem, by letting π(A) = P 0 (F ∈ A) for A ∈ G. Hence, condition (1) holds.

It remains to see that the X n can be defined on (Ω 0 , A 0 , P 0 ). To this end, it can be assumed A ₀ = σ(ξ ₁ , ξ ₂ , . . .). If P ₀ is nonatomic, it suffices to apply Corollary 5.

Suppose P ₀ has an atom A. Since A ₀ = σ(ξ ₁ , ξ ₂ , . . .), up to P ₀ -null sets, A is of the

form A = {ξ n = t n for all n ≥ 1} for some constants t n . Let σ = (σ 1 , σ 2 , . . .) be a permutation of 1, 2, . . . and A σ = {ξ n = t σ

for all n ≥ 1}. By exchangeability,

P 0 (A σ ) = P 0 (A) > 0 for all permutations σ,

and this implies t _n = t ₁ for all n ≥ 1. Let H be the union of all P ₀ -atoms. Up to P ₀ -null sets, one obtains

H ⊂ {ξ n = ξ 1 for all n ≥ 1} ⊂ {Z n = 0 for all n ≥ 1}.

If P 0 (H) = 1, thus, it suffices to let X n = 0 for all n ≥ 0. If 0 < P 0 (H) < 1, since P 0 (· | H ^c ) is nonatomic and (ξ n ) is still exchangeable under P 0 (· | H ^c ), it is not hard to define the X n on (Ω 0 , A 0 , P 0 ) in such a way that X n ∼ Z n for all n ≥ 0 and kX _n − X 0 k −→ 0. ^P

Example 10. (Pure jump processes). For each n ≥ 0, let C n = (C n,j : j ≥ 1) and Y n = (Y n,j : j ≥ 1)

be sequences of real random variables, defined on the probability space (Ω 0 , A 0 , P 0 ), such that

0 ≤ Y _n,j ≤ 1 and

∞

X

j=1

|C _n,j | < ∞.

Define

Z n (t) =

∞

X

j=1

C n,j I _{Y

_≤t} , 0 ≤ t ≤ 1, n ≥ 0.

Since Z n : (Ω 0 , A 0 ) −→ (D, B d ) is measurable, one can define µ n (·) = P 0 (Z n ∈ ·).

Then, condition (1) holds provided

C _n is independent of Y _n for every n ≥ 0,

∞

X

j=1

|C n,j − C 0,j | −→ 0 ^P

and d T V ν n,k , ν 0,k
−→ 0 for all k ≥ 1, where ν _n,k denotes the probability distribution of (Y _n,1 , . . . , Y _n,k ).

For instance, ν n,k = ν 0,k for all n and k in case Y n,j = V n+j with V 1 , V 2 , . . . a stationary sequence. Also, independence between C n and Y n can be replaced by

σ(C _n,j ) ⊂ σ(Y _n,1 , . . . , Y _n,j ) for all n ≥ 0 and j ≥ 1.

To prove (1), define Z n,k (t) = P k

j=1 C n,j I _{Y

_≤t} . For each f ∈ L,

|µ n (f ) − µ 0 (f )| ≤ |Ef (Z n ) − Ef (Z n,k )| + |Ef (Z n,k ) − Ef (Z 0,k )| + |Ef (Z 0,k ) − Ef (Z 0 )|

≤ E2 ∧ kZ n − Z n,k k + |Ef (Z n,k ) − Ef (Z 0,k )| + E2 ∧ kZ 0 − Z 0,k k

≤ E2 ∧ X

j>k

|C _n,j | + |Ef (Z n,k ) − Ef (Z _0,k )| + E2 ∧ X

j>k

|C _0,j |

where E(·) = E _P

(·). Given  > 0, take k ≥ 1 such that E2 ∧ P _j>k |C 0,j | < .

Then, lim sup

n

sup

f ∈L

|µ _n (f ) − µ ₀ (f )| < 2  + lim sup

n

sup

f ∈L

|Ef (Z _n,k ) − Ef (Z _0,k )|.

It remains to show that sup _{f ∈L} |Ef (Z n,k ) − Ef (Z 0,k )| −→ 0. Since C n is indepen- dent of Y n , up to changing (Ω 0 , A 0 , P 0 ) with some other probability space, it can be assumed

P ₀ Y _n,j 6= Y _0,j for some j ≤ k
= d T V ν _n,k , ν _0,k
;

see Proposition 1. The same is true if σ(C _n,j ) ⊂ σ(Y _n,1 , . . . , Y _n,j ) for all n and j.

Then, letting A _n,k = {Y _n,j = Y _0,j for all j ≤ k}, one obtains sup

f ∈L

|Ef (Z n,k ) − Ef (Z 0,k )| ≤ EI A

2 ∧ kZ n,k − Z 0,k k + 2 P 0 (A ^c _n,k )

≤ E2 ∧

∞

X

j=1

|C n,j − C 0,j | + 2 d T V ν n,k , ν 0,k
−→ 0.

Thus, condition (2) holds, and an application of Theorem 4 concludes the proof.

APPENDIX

Three preliminary lemmas are needed to prove Theorem 4. The first is part of the folklore about Skorohod distance, and we state it without a proof. Let

∆x(t) = x(t) − x(t−) denote the jump of x ∈ D at t ∈ (0, 1].

Lemma 11. Fix  > 0 and x _n ∈ D, n ≥ 0. Then, lim sup _n kx n − x 0 k ≤  whenever d(x _n , x ₀ ) −→ 0 and

|∆x n (t)| >  for all large n and t ∈ (0, 1) such that |∆x 0 (t)| > .

The second lemma is a consequence of Remark 6 of [5], but we give a sketch of its proof as it is basic for Theorem 4. Let µ, ν be laws on B d and F (µ, ν) the class of probabilities λ on B d ⊗ B d such that λ(· × D) = µ(·) and λ(D × ·) = ν(·). Since the map (x, y) 7→ kx − yk is measurable w.r.t. B d ⊗ B d , one can define

W u (µ, ν) = inf

λ∈F (µ,ν)

Z

1 ∧ kx − yk λ(dx, dy).

Lemma 12. For a sequence (µ _n : n ≥ 0) of probabilities on B _d , condition (1) holds if and only if W _u (µ ₀ , µ _n ) −→ 0.

Proof. The ”only if” part is trivial. Suppose W u (µ 0 , µ n ) → 0. Let Ω = D ^∞ , A = B ^∞ _d and X n : D ^∞ → D the n-th canonical projection, n ≥ 0. Take λ n ∈ F (µ 0 , µ n ) such that R 1 ∧ kx − yk λ n (dx, dy) < _n ¹ + W u (µ 0 , µ n ). Since (D, d) is Polish, λ n

admits a disintegration α n = {α n (x) : x ∈ D} (see Section 2). By Ionescu-Tulcea theorem, there is a unique probability P on B _d ^∞ such that X 0 ∼ µ 0 and

β _n (x ₀ , x ₁ , . . . , x _n−1 )(A) = α _n (x ₀ )(A), (x ₀ , x ₁ , . . . , x _n−1 ) ∈ D ⁿ , A ∈ B _d , is a regular version of the conditional distribution of X _n given (X ₀ , X ₁ , . . . , X _n−1 ) for all n ≥ 1. Under such P , one obtains (X ₀ , X _n ) ∼ λ _n (so that X _n ∼ µ _n ) and

 P kX 0 − X n k > 
≤ E P 1 ∧ kX 0 − X n k < 1

n + W u (µ 0 , µ n ) −→ 0 for all  ∈ (0, 1).

 The third lemma needs some more effort. Let φ 0 (x, ) = 0 and

φ _n+1 (x, ) = inf{t : φ _n (x, ) < t ≤ 1, |∆x(t)| > }

where n ≥ 0,  > 0, x ∈ D and inf ∅ := 1. The map x 7→ φ n (x, ) is universally measurable w.r.t. B d for all n and .

Lemma 13. Let F k be the Borel σ-field on R ^k and I ⊂ (0, 1) a dense subset. For a sequence (µ n : n ≥ 0) of probabilities on B d , condition (1) holds provided

sup

A∈F

   Z

f (x) I A φ 1 (x, ), . . . , φ k (x, )
µ n (dx) − Z

f (x) I A φ 1 (x, ), . . . , φ k (x, )
µ 0 (dx)    −→ 0 for each k ≥ 1,  ∈ I and function f : D → [−1, 1] such that |f (x) − f (y)| ≤ d(x, y)

for all x, y ∈ D.

Proof. Fix  ∈ I and write φ _n (x) instead of φ _n (x, ). As each φ _n is universally measurable w.r.t. B d , there is a set T ∈ B d such that

µ _n (T ) = 1 and I _T φ _n is B _d -measurable for all n ≥ 0.

Thus, φ n can be assumed B d -measurable for all n. Let k be such that µ 0 {x : φ r (x) 6= 1 for some r > k} < .

For such a k, define φ(x) = (φ ₁ (x), . . . , φ _k (x)), x ∈ D, and λ n (A) = µ n x : (φ(x), x) ∈ A , A ∈ F k ⊗ B d .

Since (D, d) is Polish, Proposition 2 applies to such λ _n with (F, F ) = (R ^k , F _k ) and (G, G) = (D, B _d ). Condition (b) holds by the assumption of the Lemma.

Thus, by Proposition 2, on a probability space (Ω, A, P ) there are measurable maps (Y _n , Z _n ) : (Ω, A) → (R ^k × D, F _k ⊗ B _d ) satisfying

(Y n , Z n ) ∼ λ n for all n ≥ 0, P (Y n 6= Y 0 ) −→ 0, d(Z n , Z 0 ) −→ 0. ^P Since P Y _n = φ(Z _n )
= λ n {(φ(x), x) : x ∈ D} = 1, one also obtains

(4) lim

n P φ(Z n ) = φ(Z 0 )
= 1.

Next, by (4) and d(Z _n , Z ₀ ) −→ 0, there is a subsequence (n ^P j ) such that lim sup

n

P kZ _n − Z ₀ k > 
= lim

j P kZ _n

− Z ₀ k > 
, d(Z n

, Z 0 ) −→ 0, ^a.s. P φ(Z n

) = φ(Z 0 ) for all j
> 1 − .

Define U = lim sup _j kZ n

− Z 0 k and

H = {φ r (Z 0 ) = 1 for all r > k} ∩ {φ(Z n

) = φ(Z 0 ) for all j} ∩ {d(Z n

, Z 0 ) −→ 0}.

For each ω ∈ H, Lemma 11 applies to Z ₀ (ω) and Z _n

(ω), so that U (ω) ≤ . Further, P (H ^c ) ≤ P φ r (Z 0 ) 6= 1 for some r > k
+ P φ(Z n

) 6= φ(Z 0 ) for some j

< µ ₀ {x : φ r (x) 6= 1 for some r > k} +  < 2 .

Since U ≤  on H, lim sup

n

P kZ n − Z 0 k > 
= lim

j P kZ n

− Z 0 k > 
≤ P (U ≥ )

≤ P (U = ) + P (H ^c ) < P (U = ) + 2 .

On noting that E _P 1 ∧ kZ 0 − Z _n k ≤  + P kZ n − Z ₀ k > 
, one obtains lim sup

n

W u (µ 0 , µ n ) ≤ lim sup

n

E P 1 ∧ kZ 0 − Z n k < P (U = ) + 3 .

Since I is dense in (0, 1), then P (U = ) + 3  can be made arbitrarily small for a suitable  ∈ I. Thus, lim sup _n W u (µ 0 , µ n ) = 0. An application of Lemma 12

concludes the proof. 

We are now ready for the last attack to Theorem 4.

Proof of Theorem 4. ”(1) ⇒ (2)”. Just note that

| µ n (f ) − µ ₀ (f )| = | E _P f (X n ) − E P f (X 0 ) | ≤ E P | f (X n ) − f (X ₀ )|

≤ E _P 2 ∧ kX n − X ₀ k −→ 0, for each f ∈ L, under (1).

”(2) ⇒ (1)”. Let B  = {x : |∆x(t)| =  for some t ∈ (0, 1]}. Then, B  is universally measurable w.r.t. B d and µ 0 (B  ) > 0 for at most countably many

 > 0. Hence, I = { ∈ (0, 1) : µ 0 (B  ) = 0} is dense in (0, 1).

Fix  ∈ I, k ≥ 1, and a function f : D → [−1, 1] such that |f (x) − f (y)| ≤ d(x, y) for all x, y ∈ D. By Lemma 13, for condition (1) to be true, it is enough that

lim n sup

A∈F

| µ n f I A (φ) − µ 0 f I A (φ) | = 0 (5)

where φ(x) = φ ₁ (x), . . . , φ _k (x)
, x ∈ D, and φ j (x) = φ _j (x, ) for all j.

In order to prove (5), given b ∈ (0, ^ ₂ ), define

F b = {x : |∆x(t)| / ∈ ( − 2b,  + 2b) for all t ∈ (0, 1]}, G b = {x : d(x, F b ) ≥ b 2 }.

Then,

(i) G ^c _b ⊂ F _b/2 ; (ii) φ(x) = φ(y) whenever x, y ∈ F b and kx − yk < b.

Statement (ii) is straightforward. To check (i), fix x / ∈ G b and take y ∈ F _b with d(x, y) < b/2. Let γ : [0, 1] → [0, 1] be a strictly increasing homeomorphism such that kx − y ◦ γk < b/2. For all t ∈ (0, 1],

|∆x(t)| ≤ |∆ y ◦ γ(t)| + 2 kx − y ◦ γk < |∆y(γ(t))| + b.

Similarly, |∆x(t)| > |∆y(γ(t))| − b. Since y ∈ F _b , it follows that x ∈ F _b/2 . Next, define

ψ b (x) = d(x, G b )

d(x, F _b ) + d(x, G _b ) , x ∈ D.

Then, ψ _b = 0 on G _b and ψ _b is Lipschitz w.r.t. d with Lipschitz constant 2/b. Hence, ψ b is Lipschitz w.r.t. u with Lipschitz constant 2/b (since d ≤ u). Basing on (i)-(ii) and such properties of ψ b , it is not hard to check that ψ b I A (φ) is Lipschitz w.r.t.

u, with Lipschitz constant 2/b, for every A ∈ F k . In turn, since d ≤ u and f is Lipschitz w.r.t. d with Lipschitz constant 1,

f _A = f ψ _b I _A (φ), A ∈ F _k ,

is Lipschitz w.r.t. u with Lipschitz constant (1 + 2/b). Moreover,

| µ _n f I A (φ) − µ n (f _A )| ≤ µ _n |f I _A (φ) (1 − ψ _b )| ≤ µ _n (1 − ψ _b ).

On noting that (1 + 2/b) ⁻¹ f A ∈ L for every A ∈ F k , condition (2) yields lim sup

n

sup

A∈F

| µ _n f I A (φ) − µ 0 f I A (φ) |

≤ lim sup

n

µ n (1 − ψ _b ) + sup

A∈F

|µ n (f _A ) − µ ₀ (f _A )| + µ ₀ (1 − ψ _b )

= 2 µ ₀ (1 − ψ _b ) ≤ 2 µ ₀ (F _b ^c ).

Since  ∈ I and T

b>0 F _b ^c = {x : |∆x(t)| =  for some t} = B  , one obtains lim sup

n

sup

A∈F

| µ n f I A (φ) − µ 0 f I A (φ) | ≤ 2 lim

b→0 µ ₀ (F _b ^c ) = 2 µ ₀ (B _ ) = 0.

Therefore, condition (5) holds and this concludes the proof.

 References

[1] Berti P., Rigo P. (2004) Convergence in distribution of non measurable random elements, Ann. Probab., 32, 365-379.

[2] Berti P., Pratelli L., Rigo P. (2006) Almost sure weak convergence of random probability measures, Stochastics, 78, 91-97.

[3] Berti P., Pratelli L., Rigo P. (2006) Asymptotic behaviour of the empirical process for exchangeable data, Stoch. Proc. Appl., 116, 337-344.

[4] Berti P., Pratelli L., Rigo P. (2007) Skorohod representation on a given prob- ability space, Prob. Theo. Rel. Fields, 137, 277-288.

[5] Berti P., Pratelli L., Rigo P. (2009) Skorohod representation theorem via disintegrations, Submitted, currently available at:

http://economia.unipv.it/pagp/pagine_personali/prigo/skorsankhya.pdf [6] Blackwell D., Dubins L.E. (1983) An extension of Skorohod’s almost sure rep-

resentation theorem, Proc. Amer. Math. Soc., 89, 691-692.

[7] Dudley R.M. (1999) Uniform central limit theorems, Cambridge University Press.

[8] Sethuraman J. (2002) Some extensions of the Skorohod representation theo- rem, Sankhya, 64, 884-893.

[9] van der Vaart A., Wellner J.A. (1996) Weak convergence and empirical pro- cesses, Springer.

Patrizia Berti, Dipartimento di Matematica Pura ed Applicata ”G. Vitali”, Univer- sita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy

E-mail address: berti.patrizia@unimore.it

Luca Pratelli, Accademia Navale, viale Italia 72, 57100 Livorno, Italy E-mail address: pratel@mail.dm.unipi.it

Pietro Rigo (corresponding author), Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100 Pavia, Italy

E-mail address: prigo@eco.unipv.it