We develop the use of the Zolotarev metrics on the spacesC[0,1]and D[0,1] in this context

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©Institute of Mathematical Statistics, 2015

ON A FUNCTIONAL CONTRACTION METHOD BYRALPHNEININGER ANDHENNINGSULZBACH

Goethe University Frankfurt

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the spaceC[0,1]of continuous functions endowed with uniform topology and the spaceD[0,1] of càdlàg functions with the Sko- rokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natu- ral distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach’s fixed-point theorem. We develop the use of the Zolotarev metrics on the spacesC[0,1]and D[0,1] in this context.

Applications are given, in particular, a short proof of Donsker’s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.

1. Introduction. The contraction method is an approach for proving convergence in distribution for sequences of random variables which satisfy recurrence relations in distribution. Such recurrence relations for a sequence(Yn)n≥0are often of the form

Yn d

= K r=1

Ar(n)Y^(r)

Ir⁽ⁿ⁾+b(n), n≥n0, (1)

where =^d denotes that the left-hand side and right-hand side are identically distributed, and(Y_j^(r))j≥0have the same distribution as(Yn)n≥0for allr=1, . . . , K, whereK≥1 andn0≥0 are fixed integers. Moreover,I⁽ⁿ⁾=(I₁⁽ⁿ⁾, . . . , I_K⁽ⁿ⁾)is a vector of random integers in{0, . . . , n}. The basic independence assumption that fixes the distribution of the right-hand side is that (Y_j⁽¹⁾)j≥0, . . . , (Y_j^(K))j≥0 and (A1(n), . . . , AK(n), b(n), I⁽ⁿ⁾)are independent. Note, however, that dependencies between the coefficientsAr(n),b(n)and the integersIr⁽ⁿ⁾are allowed.

Recurrences of the form (1) come up in diverse fields, for example, in the study of random trees, the probabilistic analysis of recursive algorithms, in branching

Received April 2013; revised February 2014.

MSC2010 subject classifications.Primary 60F17, 68Q25; secondary 60G18, 60C05.

Key words and phrases.Functional limit theorem, contraction method, recursive distributional equation, Zolotarev metric, Donsker’s invariance principle.

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processes, in the context of random fractals and in models from stochastic geom- etry where a recursive decomposition can be found, as well as in information and coding theory. For surveys of such occurrences, see [21,22,29]. In some applications, one may needK to depend onnor the caseK= ∞, where generalizations of the results for our case of fixedK can be stated; cf. [22], Section 4.3, for such extensions in the finite-dimensional case.

The sequence(Yn)n≥0satisfying (1) often is a sequence of real random variables with real coefficientsAr(n),b(n). However, the same recurrence appears also for sequences of random vectors (Yn)n≥0 in R^d. Then the Ar(n) are random linear maps fromR^d toR^d andb(n)is a random vector inR^d. We will also review below work that considered random sequences(Yn)n≥0into a separable Hilbert space satisfying (1) whereAr(n)become random linear operators on the space andb(n) a random vector in the Hilbert space. In the present work, we develop a limit theory for such sequences in separable Banach spaces, where our main applications are first to the spaceC[0,1]endowed with the uniform topology. Secondly, although not a Banach space, we will also be able to cover the spaceD[0,1]equipped with the Skorokhod topology. Hence, we consider sequences(Yn)n≥0of stochastic processes with state spaceRand time parameter t∈ [0,1] with continuous, respectively, cádlág paths and are interested in conditions that together with (1) allow to deduce functional limit theorems for rescaled versions of(Yn)n≥0.

For functionsf ∈C[0,1]orf ∈D[0,1], we denote the uniform norm by f∞:= sup

x∈[0,1] f (x).

For functions f, g∈D[0,1], the Skorokhod distance dsk(f, g) is used; see Sec- tion2.2.

The rescaling of the process(Yn)n≥0 can be done by centering and normaliza- tion by the order of the standard deviation in case moments of sufficient order are available. Subsequently, we assume that the scaling has already been done and we denote the scaled process by(X_n)_n≥0. Note that affine scalings of theY_n implies that the sequence(Xn)n≥0 also does satisfy a recurrence of type (1), where only the coefficients are changed:

X_n=^d K r=1

A⁽ⁿ⁾_r X^(r)

I_r⁽ⁿ⁾+b⁽ⁿ⁾, n≥n₀ (2)

with conditions on identical distributions and independence similar to recurrence (1). The coefficientsA⁽ⁿ⁾r and b⁽ⁿ⁾ in the modified recurrence (2) are typically directly computable from the original coefficientsAr(n),b(n)and the scaling used; see, for example, for the case of random vectors inR^d, [22], equation (4).

Subsequently, we consider equations of type (2) together with assumptions on the moments ofX_nwhich in applications have to be obtained by an appropriate scaling.

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For the asymptotic distributional analysis of sequences(Xn)n≥0 satisfying (2), the so-called contraction method has become a powerful tool. In the seminal paper [26], Rösler introduced this methodology for deriving a limit law for a special instant of this equation that arises in the analysis of the complexity of the Quick- sort algorithm. In the framework of the contraction method, first one derives limits of the coefficientsA⁽ⁿ⁾r ,b⁽ⁿ⁾,

A⁽ⁿ⁾_r →A_r, b⁽ⁿ⁾→b (n→ ∞) (3)

in an appropriate sense. If withn→ ∞, also theIr⁽ⁿ⁾ become large and it is plau- sible that the quantitiesXnconverge, say to a random variableX; then, by letting formallyn→ ∞, equation (2) turns into

X=^d K r=1

A_rX^(r)+b (4)

with X⁽¹⁾, . . . , X^(K) distributed as X and X⁽¹⁾, . . . , X^(K), (A1, . . . , Ak, b) independent. Hence, one can use the distributional fixed-point equation (4) to charac- terize the limit distributionL(X). The idea from Rösler [26] to formalize such an approach and to derive at least weak convergenceX_n→X consists of first using the right-hand side of (4) to define a map as follows: ifXn areB-valued random variables, denote byM(B)the space of all probability measures onBand

T:M(B)→M(B), (5)

T (μ)=L _K

r=1

A_rZ^(r)+b

, (6)

where (A1, . . . , AK, b), Z⁽¹⁾, . . . , Z^(K) are independent and Z⁽¹⁾, . . . , Z^(K) have distributionμ. Then a random variableX solves (4) if and only if its distribution L(X) is a fixed point of the mapT. To obtain fixed points ofT appropriate subspaces of M(B)are endowed with a complete metric, such that the restriction ofT becomes a contraction. Then Banach’s fixed-point theorem yields a (in the subspace) unique fixed point ofT and one can as well use the metric to also derive convergence ofL(Xn)toL(X)in this metric. If the metric is also strong enough to imply weak convergence, one has obtained the desired limit lawXn→X.

This approach has been established and applied to a couple of examples in Rösler [26,27] and Rachev and Rüschendorf [25]. In the latter paper also the flexi- bility of the approach by using various probability metrics has been demonstrated.

Later on general convergence theorems have been derived stating conditions under which convergence of the coefficients of the form (3) together with a contraction property of the map (5) implies convergence in distributionXn→X. For random variables in R with the minimal 2 metric, see Rösler [28], and Neininger [20]

forR^d with the same metric. For a more widely applicable framework for random

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variables inR^d, see Neininger and Rüschendorf [22], where in particular various problems with normal limit laws could be solved which seem to be beyond the scope of the minimalp metric; see also [23]. An extension of these theorems to continuous time, that is, to processes(X_t)_t≥0satisfying recurrences similar to (2) was given in Janson and Neininger [17].

For the case of random variables in a separable Hilbert space leading to functional limit laws, general limit theorems for recurrences (1) have been developed in Drmota, Janson and Neininger [12]. The main application there was a functional limit law for the profile of random trees which, via a certain encoding of the profile, led to random variables in the Bergman space of square integrable analytic functions on a domain in the complex plane. In Eickmeyer and Rüschendorf [13], general limit theorems for recurrences in D[0,1] under the Lp-topology were developed. Note that the uniform topology forC[0,1] and the Skorokhod topology for D[0,1] considered in the present paper are finer than the Lp-topology.

In C[0,1], the uniform topology provides more continuous functionals such as the supremum f →sup_t_∈[_0,1_]f (t) or projections f →f (s1, . . . , sk), for fixed s1, . . . , sk∈ [0,1], to which the continuous mapping theorem can be applied. In D[0,1], these functionals are also appropriate for the continuous mapping theorem if the limit random variable has continuous sample paths.

Besides the minimalpmetrics the probability metrics that have proved useful in most of the papers mentioned above is the family of Zolotarev metricsζsbeing reviewed and further developed here in Section2. All generalizations fromRvia R^d to separable Hilbert spaces are based on the fact that convergence in ζs implies weak convergence; see Section2. However, for Banach spaces this is not true in general. Counterexamples have been reported in Bentkus and Rachkauskas [4], sketched here in Section 2.1. Also completeness of the ζs metrics on appropriate subspaces of M(B) is only known for the case of separable Hilbert spaces;

see [12], Theorem 5.1.

Our study of the spaces(C[0,1], · ∞)and(D[0,1], dsk)is also based on the Zolotarev metrics ζs. Hence, we mainly have to deal with implications that can be drawn from convergence in theζs metrics as well as with the lack of knowledge about completeness ofζ_s. In Section2.3, implications of convergence in the Zolotarev metric are discussed together with additional conditions that enable to deduce in general weak convergence from convergence in ζs. A key ingredient here is a technique developed in Barbour [2] in the context of Stein’s method; see also Barbour and Janson [3]. We also obtain criteria for the uniform integrability of{Xn^s_∞|n≥0}for 0≤s≤3 in the presence of convergence in the Zolotarev metric. This enables in applications as well to obtain moments convergence of the sup-functional.

In Section3, we give general convergence theorems in the framework of the contraction method first for a general separable Banach space and then apply and refine this to the space(C[0,1],·_∞)and develop a technique to also apply this to the metric space(D[0,1], dsk). In particular, based on Janson and Kaijser [16], we

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give a criterion for the finiteness of the Zolotarev metric on appropriate subspaces that can easily be checked in applications.

To compensate for the lack of knowledge about completeness of the ζs metrics, we need to assume that the mapT in (5) has a fixed point in an appropriate subspace of M(C[0,1]) andM(D[0,1]), respectively. In applications, one may verify this existence of a fixed point either by guessing one successfully: in the application of our framework to Donsker’s functional limit theorem in Section4.1, the Wiener measure can easily be guessed and be seen to be the fixed point of the map T coming up there. Alternatively, in general the existence of a fixed point may arise from infinite iteration of the mapT: applied to some probability measure, such an iteration has a series representation for which one may be able to show that it is the desired fixed point. This path is being taken in an application of our framework outlined in Section4.2.

In Section 4.1, we apply our functional contraction method to derive a short proof of Donsker’s functional limit theorem. This does not require the full generality of our setting but illustrates how self-similarities can easily been exploited with this approach. The application in Section4.2is on the asymptotic study of fundamental complexities in computer science. Here, the full generality of our approach is needed to obtain a functional limit law. We highlight and discuss the use of our conditions (C1)–(C5) formulated in Section3on the recurrence (2) at this example. Details on the verification of the conditions are contained in Broutin, Neininger and Sulzbach [6] where, based on the functional limit law, also various long open standing problems on the complexities in computer science are solved.

2. The Zolotarev metric. Let (B, · ) be a real Banach space and B its Borelσ-algebra. In Section2.1, we assume that the norm onB induces a separable topology. We denote byM(B)the set of all probability measures on(B,B).

First, we introduce the Zolotarev metricζs and collect some of its basic properties, mainly covered in [32,33]. In the second subsection, we define our use of the Zolotarev metrics on the metric space(D[0,1], dsk). Although not a Banach space, we will be able to declare the Zolotarev metricsζs on(D[0,1], dsk)using the no- tion of differentiability of functionsD[0,1] →Rinduced by the supremum norm onD[0,1]. We also comment in Remarks6and7on delicate measurability issues for the nonseparable Banach space(B, · )=(D[0,1], · ∞)and the realm of our methodology when working with the coarser (separable) topology onD[0,1] induced by the Skorokhod metric. In the third subsection, conditions that allow to conclude from convergence inζ_s to weak convergence are studied for the case (B, · )=(C[0,1], · _∞)as well as for the case(D[0,1], dsk). We also discuss further implications fromζs-convergence in these two spaces as well as criteria for finiteness ofζs. Additional material to the content of this section can be found in the second author’s dissertation [31], Chapter 2.

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2.1. Definition and basic properties. For functions f:B →R, which are Fréchet differentiable, the derivative off at a pointx is denoted byDf (x). Note thatDf (x)is an element of the spaceL(B,R) of continuous linear forms onB. We also consider higher order derivatives, whereD^mf (x)denotes themth derivative off at a pointx. Thus,D^mf (x)is a continuousm-linear (or multilinear) form onB. The space of continuous multilinear formsg:B^m→Ris equipped with the norm

g = sup

h₁≤1,...,h_m≤1

g(h1, . . . , hm).

For a comprehensive account on differentiability in Banach spaces, we refer to Cartan [7]. Subsequently,s >0 is fixed and form:= s −1 andα:=s−mwe define

Fs=f:B→R:D^mf (x)−D^mf (y)≤ x−y^α, ∀x, y∈B. (7)

Forμ, ν∈M(B), the Zolotarev distance betweenμandνis defined by ζ_s(μ, ν)= sup

f∈Fs

E f (X)−f (Y ), (8)

where X and Y are B-valued random variables with L(X)=μ and L(Y )=ν.

Here, L(X) denotes the distribution of the random variable X. The expres- sion in (8) does not need to be finite or even well defined. However, we have ζs(μ, ν) <∞if

x^sdμ(x),

x^sdν(x) <∞ (9)

and

f (x, . . . , x) dμ(x)= f (x, . . . , x) dν(x) (10)

for any boundedk-linear formf onB and any 1≤k≤m. For random variables X, Y in B, we use the abbreviation ζs(X, Y ):=ζs(L(X),L(Y )). Finiteness of ζs(X, Y )inR^dfails to hold ifXandY do not have the same mixed moments up to orderm. The assumption on the finite absolute moment of orders can be relaxed slightly; see Theorem 4 in [34].

We denote

Ms(B):=μ∈M(B)

x^sdμ(x) <∞ and for allν∈Ms(B)denote

Ms(ν):=μ∈Ms(B)|μandνsatisfy (10).

Thenζs is a metric on the spaceMs(ν)for anyν∈Ms(B); see [35], Remark 1, page 198.

A crucial property ofζsin the context of recursive decompositions of stochastic processes is the following lemma; see Theorem 3 in [34]. A short proof is given for the reader’s convenience.

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LEMMA1. LetBbe a Banach space andg:B→Ba linear and continuous operator.Then we have

ζs

g(X), g(Y )≤ g^sζs(X, Y ), L(X),L(Y )∈Ms(ν).

Here,gdenotes the operator norm ofg,that is,g =sup_x_∈_B,_x_≤₁g(x). PROOF. Note thatgis also bounded. It suffices to show that

g⁻^sf ◦g:f ∈Fs

⊆Fs,

where F_s is defined analogously to Fs in B. Let f ∈Fs and η:= g⁻^sf ◦ g. Then η is m-times continuously differentiable and we have D^mη(x) = g⁻^s(D^m(f (g(x)))◦g^⊗^mforx∈B. Here,g^⊗^m:B^m→(B)^mdenotes the map- pingg^⊗m(h1, . . . , hm)=(g(h1), . . . , g(hm)). This implies

D^mη(x)−D^mη(y)= g⁻^sD^mfg(x)◦g^⊗^m−D^mfg(y)◦g^⊗^m

≤ g^−αg(x)−g(y)^α

= g⁻^αg(x−y)^α≤ x−y^α. The assertion follows.

Another basic property is thatζsis(s,+)ideal.

LEMMA 2. The metricζ_s is ideal of orders on Ms(ν)for anyν∈Ms(B), that is,we have

ζs(cX, cY )= |c|^sζs(X, Y ), ζs(X+Z, Y +Z)≤ζs(X, Y )

for anyc∈R\ {0},L(X),L(Y )∈Ms(ν)and random variablesZinB,such that (X, Y )andZare independent.

The lemma directly implies

ζ_s(X₁+X₂, Y₁+Y₂)≤ζ_s(X₁, Y₁)+ζ_s(X₂, Y₂) (11)

for L(X1),L(Y1)∈Ms(ν1)and L(X2),L(Y2)∈Ms(ν2)with arbitrary ν1, ν2∈ Ms(B)such that(X1, Y1)and(X2, Y2)are independent.

We want to give a result similar to Lemma 1 where the linear operator may also be random itself. We focus on the case thatBeither equalsB orRwhere an extension toR^d ford >1 is straightforward. LetB^∗be the topological dual ofB andBbe the space of all continuous linear maps fromB toB. Endowed with the operator norms

fop= sup

x∈B,x≤1

f (x), fop= sup

x∈B,x≤1

f (x),

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both spaces, B^∗ andB, respectively, are Banach spaces. However, these spaces are typically nonseparable, hence not suitable for our purposes of measurability.

Therefore, we will equip them with smallerσ-algebras. Similar to the use of weak-

* convergence, letB^∗ be theσ-algebra onB^∗ that is generated by all continuous (with respect to · op) linear forms ϕ on B^∗ (i.e., elements of the bidual B^∗∗) of the form ϕ(a)=a(x) for some x ∈B. Note that the set of these continuous linear forms coincides with the bidualB^∗∗if and only ifB is reflexive, a property that is not satisfied in our applications. We move on toBand defineBto be the σ-algebra generated by all continuous (with respect to · op) linear mapsψfrom Bto B of the form ψ (a)=a(x) for some x∈B. By Pettis’ theorem, we have B=σ (∈B^∗). Hence, ifS⊆B^∗withB=σ (∈S), thenBis also generated by the continuous linear formsonBthat can be written as(a)=(a(x))for∈S andx∈B.

Using the separability of B, it is now easy to see that the norm-functionals B^∗→R, f → fop and B→R, f → fop are B^∗–B(R) measurable and B–B(R)measurable, respectively.

DEFINITION 3. By a random continuous linear form on B, we denote any random variable with values in(B^∗,B^∗). Analogously,random continuous linear operators onB are random variables with values in(B, B).

Note that the definition of the σ-algebras B^∗ and Bimplies in particular that for anya∈B^∗ora∈B, x ∈B, random continuous linear form or operatorAand random variableXinB, we have that the compositionsa(X),A(x)andA(X)are again random variables. The latter property follows from measurability of the map (a, x)→a(x)with respect to(B^∗⊗B)–B(R)and(B⊗B)–B, respectively. In the case of the dual space, this follows as for anyr∈Rwe have

(a, x)∈B^∗×B:a(x) < r

=

k≥1 m≥1

n≥m

i≥1

a∈B^∗:a(ei) < r−1/k× {x∈B:x−ei<1/n},

where{ei|i≥1}denotes a countable dense subset of B; the caseBbeing analo- gous.

The following lemma follows from Lemma1by conditioning.

LEMMA 4. Let L(X),L(Y )∈Ms(ν) for some ν∈Ms(B). Then, for any random linear continuous form or operatorAwithE[A^sop]<∞independent of XandY,we have

ζs

A(X), A(Y )≤EA^s_opζs(X, Y ).

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Zolotarev gave upper and lower bounds forζs, most of them being valid if more structure onBis assumed. Subsequently, only an upper bound in terms of the min- imal_pmetric is needed. Forp >0 andμ, ν∈Mp(B), the minimal_pdistance betweenμandνis defined by

p(μ, ν)=infE X−Y^p^(1/p)^∧¹,

where the infimum is taken over all common distributionsL(X, Y )with marginals L(X)=μandL(Y )=ν. We abbreviatep(X, Y ):=p(L(X),L(Y )).

The next lemma gives an upper bound ofζs in terms ofs where the first statement follows from the Kantorovich–Rubinstein theorem and the second essentially coincides with Lemma 5.7 in [12].

LEMMA5. LetL(X),L(Y )∈Ms(ν)for someν∈Ms(B)withB separable.

Ifs≤1then

ζs(X, Y )=s(X, Y ).

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Ifs >1then

ζ_s(X, Y )≤E X^s¹⁻^1/s+E Y^s¹⁻^1/s_s(X, Y ).

IfXn, Xare real-valued random variables,n≥1, thenζs(Xn, X)→0 implies convergence of absolute moments of order up tossince there is a constantCs>0 such that the functionx→Cs|x|^s is an element ofFs, hence|E[|Xn|^s− |X|^s]| ≤ C_s⁻¹ζs(Xn, X).

We proceed with the fundamental question of how convergence in theζs distance relates to weak convergence on B. By the first statement of the previous lemma, or more elementary, by the proof of the Portmanteau lemma [5], Theo- rem 2.1(ii)–(iii), one obtains that for 0< s≤1 convergence in theζs metric implies weak convergence; see also [12], page 300.

IfB is a separable Hilbert space, then for anys >0 convergence in theζs metric implies weak convergence. This was first proved by Giné and León in [15], see also Theorem 5.1 in [12]. In infinite-dimensional Banach spaces convergence in theζs metric does not need to imply weak convergence: for any probability distri- butionμonB=C[0,1]with zero mean andx^s_∞dμ(s) <∞for somes >2, that is pre-Gaussian, that is, there exists a Gaussian measure ν on C[0,1] with zero mean and the same covariance as μ, one has ζs-convergence of a rescaled sum of independent random variables with distributionμ towardν; see inequality (48) in [32]. However, pre-Gaussian probability distributions supported by a bounded subset ofC[0,1]that do not satisfy the central limit theorem can be found in [30]. For the central limit theorem in Banach spaces, see [18]. Note that convergence with respect to ζs implies convergence of the characteristic functions, hence ζ_s(X_n, X)→0 implies thatL(X)is the only possible accumulation point of(L(Xn))n≥0in the weak topology.

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2.2. The Zolotarev metric on(D[0,1], dsk). In this section, we discuss our use of the Zolotarev metric on the metric space(D[0,1], dsk) of càdlàg functions on [0,1]endowed with the Skorokhod metric defined by

dsk(f, g)

=infε >0|maxf (t)−gτ (t),τ (t)−t< εfor allt∈ [0,1] for some monotonically increasing and bijectiveτ:[0,1] → [0,1]. The Borelσ-algebra of the induced topology is denoted byBsk. For a general introduction to this space, see Billingsley [5], Chapter 3. In particular,(D[0,1], dsk)is a Polish space,Bskcoincides with theσ-algebra generated by the finite-dimensional projections, theσ-algebra generated by the open spheres (with respect to the uniform metric) and theσ-algebra generated by all norm-continuous linear forms on D[0,1]; see [24], Theorem 3. Subsequently, norm onD[0,1] will always refer to the uniform norm · ∞. Moreover, the norm functionD[0,1] →R,f → f∞

is Bsk–B(R) measurable. By Theorem 2, respectively, Theorem 4, in [24], any norm-continuous linear form onD[0,1]is Bsk–B(R)measurable and any norm- continuous linear map fromD[0,1] toD[0,1] isBsk–Bsk measurable. Recently, Janson and Kaijser [16], Theorem 15.8, generalized the latter result and proved that any norm-continuousk-linear form onD[0,1]is(Bsk)^⊗^k–B(R)measurable.

We do, however, not know whetherFs defined in (7) based on the uniform norm onD[0,1]is a subset of theBsk–B(R)measurable functions. Hence, we denote the Bsk–B(R)measurable functions byEand define the Zolotarev metrics analogously to (8) by

ζs(μ, ν)= sup

f∈Fs∩E

E f (X)−f (Y ),

where X and Y are (D[0,1], dsk)-valued random variables with L(X)=μ and L(Y )=ν.

We denote byMs(D[0,1])the set of probability distributionsμonD[0,1]with x^s_∞dμ(x) <∞and forν∈Ms(D[0,1]), we defineMs(ν)to be the subset of measuresμfromMs(D[0,1])satisfying (10). Thenζsis a metric onMs(ν)for all ν∈Ms(D[0,1]), Lemmas1and2, inequality (11), Lemma5where (12) is to be replaced byζ_s(X, Y )≤_s(X, Y ), and the implicationζ_s(X_n, X)→0⇒X_n→X in distribution if 0< s≤1 remain valid.

The situation becomes more involved concerning random linear forms and operators as defined in Definition 3 in the separable Banach case. Let D[0,1]^∗ and D[0,1] be the dual space, respectively, the space of norm-continuous endomorphisms on D[0,1] as in the Banach case. For reasons of measurability, we need to restrict to smaller subspaces. Let D[0,1]^∗c ⊆D[0,1]^∗ be the subset of functions that are additionally continuous with respect to dsk. Analogously, D[0,1]c ⊆D[0,1] are those endomorphism which are continuous regarded as maps from(D[0,1], dsk)to(D[0,1], dsk). We endowD[0,1]^∗cwith theσ-algebra

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generated by the functionf → fopand all elementsϕ ofD[0,1]^∗∗of the form ϕ(a)=a(x)for somex∈D[0,1]. Also theσ-algebra onD[0,1]cis generated by the functionf → fop and the continuous linear mapsψ:D[0,1] →D[0,1]of the formϕ(a)=a(x) for somex∈D[0,1]. Under these conditions, we have the same measurability results as in the Banach case and Lemma4remains valid.

REMARK6. Note that we could as well develop the use of the Zolotarev metric together with the contraction method for the Banach space(D[0,1], · _∞).

This can be done analogously to the discussion of Sections 2.3 and 3 and in fact would lead to a proof of Donsker’s theorem similar to the one given in Sec- tion4.1.1 when replacing the linear interpolationSⁿ=(S_tⁿ)t∈[0,1] by a constant (càdlàg) interpolation of the random walk. However, the applicability of such a framework seems to be limited due to measurability problems in the nonseparable space(D[0,1], · ∞): for example, the random functionXdefined by

Xt =1_{t≥U}, t∈ [0,1]

withU being uniformly distributed on the unit interval is known to be nonmeasur- able with respect to the Borel-σ-algebra on (D[0,1], · ∞). However, we have applications of the functional contraction method developed here in mind on processes with jumps at random times. A typical example in the context of random trees is given in Section4.2; see also [6]. Hence, in order to even have measurability of the processes considered it requires to work with the coarser Skorokhod topology than the uniform topology and this is our reason for using the Zolotarev metric on(D[0,1], dsk)instead of(D[0,1], · ∞).

REMARK 7. Although the methodology developed below covers sequences (Xn)n≥0of processes with jumps at random times these times will typically need to be the same for all n≥n0. In particular, sequences of processes with jumps at random times that require a (uniformly small) deformation of the time scale to be aligned cannot be covered by this methodology. The technical reason is that in condition (C1) below (see Section3) the convergence of the random continuous endomorphisms A⁽ⁿ⁾r −Ars is with respect to the operator norm based on the uniform norm which in general does not allow a deformation of the time scale.

2.3. Weak convergence on(C[0,1], · ∞)and(D[0,1], dsk). In this subsection, we only consider the spaces(C[0,1], · ∞)and(D[0,1], dsk).

For random variables X=(X(t))t∈[0,1], Y =(Y (t))t∈[0,1] in (C[0,1], · _∞) withζs(X, Y ) <∞we have

ζs

X(t1), . . . , X(tk),Y (t1), . . . , Y (tk)≤k^s/2ζs(X, Y ) (13)

for all 0≤t₁≤ · · · ≤t_k≤1. This follows from Lemma 1 using the continuous and linear functiong:C[0,1] →R^k, g(f )=(f (t1), . . . , f (tk))and observing that

(12)

g =√

k. The bound ζs((X(t1), . . . , X(tk)), (Y (t1), . . . , Y (tk)))≤ζs(X, Y )can be obtained ifR^k is endowed with the max-norm instead of the Euclidean norm.

However, no use of this is made here. Hence, we obtain for random variablesXn, Xin(C[0,1], · ∞),n≥1, the implication

ζ_s(X_n, X)→0 ⇒ X_n^f.d.d.−→X.

Here, −→^f.d.d. denotes weak convergence of all finite-dimensional marginals of the processes. Additionally, ifZ is a random variable in[0,1], independent of(Xn) andX, then applying Lemma4with the random continuous linear formAdefined byA(f )=f (Z)implies

ζ_sXn(Z), X(Z)≤E Z^sζs(Xn, X).

(14)

In the càdlàg case, that is,X =(X(t))t∈[0,1], Y =(Y (t))t∈[0,1] being random variables in(D[0,1], dsk)inequality (13) remains true by Lemma1. (The fact that gis not continuous with respect to the product Skorokhod topology does not cause problems since measurability is sufficient here.) Next, in general, the operatorA is no element ofD[0,1]^∗c. Hence, we cannot apply Lemma4to deduce (14). Nev- ertheless, by Theorem 2 in [34], the convergence of the characteristic functions of X_n(t) is uniform in t, hence we also have convergence in distribution of X_n(Z) toX(Z). The same argument works for the moments of Xn(Z). We summarize these properties in the following proposition, where−→^d denotes convergence in distribution.

PROPOSITION 8. For random variables Xn, X in (C[0,1], · ∞) or (D[0,1], dsk),n≥1,withζs(Xn, X)→0forn→ ∞we have

Xn f.d.d.

−→X.

L(X)is the only possible accumulation point of(L(Xn))n≥1in the weak topology.

For allt∈ [0,1]we have

Xn(t)−→^d X(t), E Xn(t)^s→E X(t)^s.

For any random variableZ in[0,1]being independent of(X_n)andX,we have E Xn(Z)^s→E X(Z)^s, Xn(Z)−→^d X(Z).

To conclude from convergence in the ζs metric to weak convergence on (C[0,1], · ∞)or(D[0,1], d_sk), further assumptions are needed. Let, forr >0,

Cr[0,1] :=f ∈C[0,1]|∃0=t1< t2<· · ·< t=1, ∀i=1, . . . , : (15) |ti−ti−1| ≥r, f|[t_i−1,t_i]is linear

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denote the set of all continuous functions for which there is a decomposition of [0,1]into intervals of length at leastrsuch that the function is piecewise linear on those intervals. Analogously, we define

Dr[0,1] :=f ∈D[0,1]|∃0=t1< t2<· · ·< t=1, ∀i=1, . . . , : (16) |ti−ti−1| ≥r, f|_[t_i−1,t_i)is constant, continuous in 1.

THEOREM9. LetX_nbe random variables inCrn[0,1],n≥0,andXa random variable inC[0,1].Assume that for0< s≤3withs=m+αas in(7)

ζs(Xn, X)=o

log⁻^m 1

rn

. (17)

ThenX_n→X in distribution.The assertion remains valid ifC[0,1],Crn[0,1]are replaced byD[0,1],Dr_n[0,1] endowed with the Skorokhod topology and X has continuous sample paths.

As discussed above, ζs convergence does not imply weak convergence in the spaces C[0,1] and D[0,1] without any further assumption such as (17). In the counterexample from [30], the sequence Sn/√

n there converges to a Gaussian limit with respect to ζs for 2< s ≤ 3 where the rate of convergence is upper bounded by the ordern¹⁻^s/2; see [32] or [31]. Moreover, the sequence is piecewise linear but the sequencer_ncan only be chosen of the order(cn)⁻²ⁿfor somec >0.

Hence, (17) is not satisfied.

In applications such as our proof of Donsker’s functional limit law in Sec- tion 4.1.1or the application of the present methodology to a problem from the probabilistic analysis of algorithms in [6], the rate of convergence will typically be of polynomial order which is fairly sufficient.

We postpone the proof of the theorem to the end of this section and state two variants, where the first one, Corollary 10, contains a slight relaxation of the assumptions that is useful in applications such as in the analysis of the complexity of partial match queries in quadtrees; see Section4.2or [6]. The second one will be needed in the cases >2; see Section4.1.

COROLLARY 10. LetXn, X beC[0,1] valued random variables,n≥0,and 0< s≤3withs=m+α as in(7).SupposeXn=Yn+hn withYnbeingC[0,1] valued random variables andhn∈C[0,1],n≥0,such thathn−h_∞→0for a h∈C[0,1]and

PYn∈/Cr_n[0,1]→0.

(18) If

ζs(Xn, X)=o

log⁻^m 1

rn

,

(14)

then

Xn

−→d X.

The statement remains true if C[0,1] andCrn[0,1] are replaced by D[0,1] and Dr_n[0,1] endowed with the Skorokhod topology, respectively, X has continuous sample paths andhremains continuous.

COROLLARY 11. Let X_n, Y_n, X beC[0,1] valued random variables, n≥0, and 0< s≤3 with s =m+α as in (7).Suppose X_n∈Crn[0,1] for all n and Yn→Xin distribution.If

ζs(Xn, Yn)=o

log^−m 1

rn

, then

Xn

−→d X.

The statement remains true if C[0,1] andCr_n[0,1] are replaced by D[0,1] and Dr_n[0,1]endowed with the Skorokhod topology,respectively,andX has continu- ous sample paths.

InC[0,1](orD[0,1], if the limitXhas continuous paths), convergence in distribution implies distributional convergence of the supremum normXn∞by the continuous mapping theorem. In applications, one is also interested in convergence of moments of the supremum. For random variablesX inC[0,1] orD[0,1], we denote by

Xs:=E X^s_∞^(1/s)^∧¹ theL_s-norm of the supremum norm.

THEOREM12. LetXn, X beC[0,1]valued random variables and0< s≤3 withXns,Xs <∞for alln≥0. Suppose one of the following conditions is satisfied:

(1) X_n∈Crn[0,1]for allnand ζs(Xn, X)=o

log^−m

1 r_n

. (19)

(2) Xn=Yn+hn with Yn being C[0,1] valued random variables and hn ∈ C[0,1],n≥0,such thathn−h∞→0for ah∈C[0,1],

E Xn^s_∞1_{Y_n∈/Crn[0,1]}

→0 (20)

and

ζ_s(X_n, X)=o

log⁻^m 1

rn

.

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(3) (Yn)n≥0 is a sequence ofC[0,1] valued random variables withYn≤Z al- most surely for aC[0,1]valued random variableZwithZs<∞,Xn∈Cr_n[0,1] for allnand

ζ_s(X_n, Y_n)=o

log⁻^m 1

rn

.

Then {Xn^s_∞|n ≥ 0} is uniformly integrable. All statements remain true if C[0,1],Cr_n[0,1]are replaced byD[0,1],Dr_n[0,1]andhin item(2)remains con- tinuous.

It is of interest whether the metric space(Ms(ν), ζs)is complete. This is true for 0< s≤1. Also, in the case thatBis a separable Hilbert space, this holds true;

see Theorem 5.1 in [12]. Nevertheless, the problem remains open in the general case, in particular in the casesC[0,1]andD[0,1] withs >1. We can only state the following proposition.

PROPOSITION13. LetB=(C[0,1], · ∞)orB=(D[0,1], dsk),s >0and ν∈Ms(B).Furthermore,let(μn)n≥0be a sequence of probability measures from Ms(ν)which is a Cauchy sequence with respect to theζsmetric.Then there exists a probability measureμonR^[^0,1^]such that,asn→ ∞,

μn f.d.d.

−→μ.

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PROOF. Let L(Xn)= μn for all n ≥ 0. According to (13), (Xn(t1), . . . , Xn(tk))n≥0 is a Cauchy sequence and hence it exists a random variable Yt₁,...,t_k

inR^kwith

Xn(t1), . . . , Xn(tk)−→^d Yt₁,...,t_k (n→ ∞).

The set of distributions ofYt₁,...,t_kfor 0≤t1<· · ·< tk≤1 andk∈Nis consistent so there exists a process Y on the product space R^[^0,1^] whose distribution satis- fies (21).

REMARK14. If the distributionμfound in Proposition13has a version with continuous paths then condition (10) forμnandμis satisfied.

We now present proofs of the theorems and corollaries of the present sections.

Theorem9essentially follows directly from Theorem 2 in [2]; see also [3]. Nev- ertheless, we present a version of the proof given there so that we can deduce the variants and implications given in our other statements. A basic tool are Theo- rems 2.2, 2.3 and 2.4 in Billingsley [5].

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LEMMA 15. Let (μn)n≥0, μ be probability measures on a separable met- ric space (S, d). For r >0, x ∈S let Br(x)= {y ∈S:d(x, y) < r}. If for any x1, . . . , xk∈S, γ1, . . . , γk>0withμ(∂Bγ_i(xi))=0fori=1, . . . , kit holds

μ_n

i∈I

B_γ_i(x_i)

→μ

i∈I

B_γ_i(x_i)

, (22)

whereI= {1, . . . , k},thenμ_n→μweakly.

Let (S, d) =(D[0,1], dsk). Then the assertion remains true when the balls B_γ_i(x_i)are still defined with respect to theuniformdistance andμ(C[0,1])=1.

PROOF. The first part of the lemma is a special case of Theorem 2.4 in [5].

To prove the assertion in the càdlàg space, we apply Theorem 2.2 in [5] upon choosingAP there to be the set of finite intersection of setsAwhereAis either a μ-continuous open sphere (in the uniform distance) whose center lies inC[0,1]or a measurable set with positive uniform distance fromC[0,1]. Using (22) and the inclusion-exclusion formula, it is easy to see thatμn(C)→0 for any measurable set C with positive uniform distance from C[0,1], in particular μ_n(A)→μ(A) for anyA∈AP. Moreover, we can decompose any open setO∈D[0,1] (in the Skorokhod topology) intoO andO\O with

O:=

x,δ

B_x^·(δ),

where the union is over all x ∈O ∩C for a countable set C that is dense in C[0,1] andδ∈Q⁺such thatBx^·(δ)⊆O andBx^·(δ)isμ-continuous. We have O∩C[0,1] ⊆Osince any ball in the metricdskwith center inC[0,1]contains a concentric ball in the uniform distance. Hence,

O\O=

δ∈Q⁺

x∈O\O:y−x> δfor ally∈C[0,1].

Thus, any open setO is a countable union of sets inAP which proves all conditions of Theorem 2.2 in [5] to be satisfied and the claim follows.

A main difficulty in deducing weak convergence from convergence inζ_s com- pared to the Hilbert space case is the nondifferentiability of the norm function x→ x∞; see [10], page 147. We will instead use the smootherL_p-norm which approximates the supremum norm in the sense that

Lp(x)→ x∞

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for any fixedx∈C[0,1]asp→ ∞.

For the remaining part of this section,p, for fixed values or tending to infinity, is always to be understood as an even integer withp≥4. We use the Bachmann–

Landau big-Onotation.