Limits of P´olya urns with innovations

We consider a version of the classical P´olya urn scheme which incorporates innovations. The space S of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns C balls of the same color and additional balls of diﬀerent colors given by some ﬁnite point process ξ on S , where the distribution P s of the pair ( C,ξ ) depends on the sampled color s . We suppose that the average number of copies E s ( C ) is the same for all s ∈ S , and that the intensity measures of innovations have the form E s ( ξ ) = a ( s ) µ for some ﬁnite measure µ and a modulation function a on S that is bounded away from 0 and ∞ . We then show that the empirical distribution of the colors in the urn converges to the normalized intensity µ . In turn, diﬀerent regimes for the ﬂuctuations are observed, depending on whether E s ( C ) is larger or smaller than µ ( a ).


Introduction
At each step of an original Pólya urn scheme, a ball is drawn uniformly at random from an urn, independently of the preceding steps.One observes its color; then the ball is returned in the urn together with an additional ball of the same color.More generally, one can consider random replacement schemes, for which one rather returns random numbers of balls of different colors according to some fixed distribution that depends only on the color of the sampled ball.The quantities of interest are the proportions of balls with given colors after a large number of steps.Most often, it is assumed that the set of all possible colors is finite, with the notable exception of certain recent contributions which will be discussed at the end of this Introduction.General Pólya urn schemes are used as models in a variety of fields including Computer Sciences, Biology, Social Sciences, etc.; see e.g.[19] and references therein.
We are interested in a variation of the Pólya urn scheme which incorporates innovations, in the sense that at each step, balls with new colors that have never been used before can be returned in the urn, and the space of colors is an arbitrary measurable space.Such an urn scheme with innovation has been first introduced 1 in 1955 by Herbert Simon [23] in terms of a simple model for producing a text by random iterations.One starts with a first word, say w 1 , and when one has already written a text of length k ≥ 1, say w 1 . . .w k , one appends the next word w k+1 which is either, with fixed probability p, a repetition of w u(k) , where u(k) has the uniform distribution on {1, . . ., k}, or, with complementary probability 1 − p, a new word different from all the preceding.Further situations where innovations or novelties are combined with urn schemes have been considered by [10,24,11], amongst others.Needless to recall, the association of stochastic reinforcement and innovation is an important aspect of machine learning on which there exists of course a huge literature.
In a recent work [8] that was partly motivated by the study of certain step reinforced random walks, we analyzed the effects of linear reinforcement on the sequence of empirical distributions in Simon model.It was observed there that when one takes for the different words independent uniform random variables on [0, 1], the Glivenko-Cantelli theorem still holds, that is there is the almost-sure convergence of the empirical distribution functions of the reinforced sequence towards the identity function on [0, 1], but the Donsker theorem is affected.Indeed, the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when the repetition probability p < 1/2; a further rescaling is needed when p > 1/2 and the limit is then a bridge with exchangeable increments and discontinuous paths.
Here, we consider more generally a Borel space S as the set of possible colors of balls.We write L ∞ for the space of bounded measurable functions f : S → R, and use often use the short notation for a finite measure ν on S and f ∈ L ∞ .In particular, the total mass of ν is denoted by ν (1), where 1 stands for the function identical to 1 on S. Whenever ν ≡ 0, we write ν = ν(1) −1 ν for the normalized probability measure associated to ν.
A typical replacement consists of a pair (C, ξ), where C is a random variable with values in {−1, 0, 1, 2, . ..} which represents the number of copies of the sampled ball which are returned in the urn (the case C = −1 accounts for the situation where the sampled ball is removed from the urn), and ξ a point process on S which represents the random family of new balls which are simultaneously added.The dynamics are hence fully encoded by the kernel of laws (P s ) s∈S that specify the distributions of the pair (C, ξ) as a function of the color s of the sampled ball.We implicitly assume that the probability kernel s → P s is measurable, and to avoid discussing situations where the urn could be emptied, we always assume henceforth that for all s ∈ S, where, by the preceding notation, ξ(1) accounts for the number of innovative balls.
More precisely, at each step, independently of the previous steps, we sample a ball uniformly at random from the urn and observe its color, say s, as well as a pair (C, ξ) with law P s .Given C = k and ξ = j i=1 δ si , with δ s denoting the Dirac measure at s, we then return that ball in the urn together with k balls with the same color s and j further balls with colors s 1 , . . ., s j (it is not required that the s i are all distinct).We represent the composition of the urn after n steps by a counting measure U n on S such that the atoms of U n are the colors of the balls present in the urn after n steps, and their multiplicities the number of balls of these colors.Then for f ∈ L ∞ , U n (f ) is the sum of the f (s) for the colors s ∈ S present in the urn after n steps and repeated according to their multiplicities.
We now introduce the three crucial assumptions of our study.First, we suppose that the average number of copies that are returned at a typical step does not depend on the sampled color, viz.
where the notation E s is used for the mathematical expectation under P s .Second, we suppose that the intensity measures of innovations are all proportional to some fixed measure µ on the space of colors.Specifically, we assume that there exists a measurable function a on S that is bounded away from 0 and from ∞, viz.with such that for every s ∈ S, the intensity measure of the point processes ξ under P s is given by We should think of a as a factor which modulates the intensity of innovations as a function of the color of the sampled ball.Last, we shall also need an assumption of uniform upper and lower boundedness for the total number of balls which are returned at each step, namely that 0 < inf s∈S P s (C + ξ(1) ≥ 1) and sup We then introduce two fundamental constants, As the notation suggests, these two quantities shall be viewed as eigenvalues.More precisely, consider the operator R : which describes averaged replacements as a function of the color of the sampled ball.Then λ 1 and λ 2 arise as the two eigenvalues of R; specifically and also Observe from (1) that and also that λ 1 > λ 2 , since µ(a) > 0.
When the set of colors S is finite, say S = {1, . . ., d}, the present model with innovation merely rephrases the classical setting of urns with random replacements schemes and the average replacement operator R is given by a d × d matrix.The requirements (2) and (3) entail that the latter has the form R = λ 2 Id + R 1 , with R 1 a matrix of rank 1.In this situation, the spectral properties of R observed above are immediate, and the results in this work should not come much as a surprise.Nonetheless dealing with a general space of colors cannot be reduced (e.g. by approximations) to the case when S is finite.Although our analysis will of course borrow the same guiding lines as for the classical setting, in particular spectral analysis of the mean replacement operator and martingale limit theorems, new ideas are also needed to resolve several issues.
An important result for urn schemes with finitely many colors is that the first order asymptotic of the contain of the urn as the number of steps goes to infinity is determined by the largest eigenvalue of the mean replacement matrix and its eigenvectors.The same feature holds in the present setting; it implies the almost-sure convergence of the empirical distribution of colors to the normalised intensity of innovation.
Theorem 1. Assume (2), (3), and (4).For every f ∈ L ∞ , we have where µ := µ(1) −1 µ is the normalised intensity of innovation.As a consequence, we have also in terms of the empirical distribution of colors that We next turn our attention to the fluctuations of the empirical distributions; in order to state our main result, we need to introduce first a few more objects.For any f ∈ L ∞ , we set which defines a positive semidefinite quadratic form on L ∞ .We denote by G = (G(f ) : f ∈ L ∞ ) the associated Gaussian process, i.e.G is centered with covariance In general, the variables G(f ) and G(g) are not always independent when f and g are orthogonal in L 2 (µ), due the component E(ξ(f )ξ(g)) in the covariance.We also introduce the Gaussian bridge where Plainly, the Gaussian bridge is translation invariant in the sense that G (br) (f ) = G (br) (f + c) for any c ∈ R, and coincides with G on the hyperplane of functions f ∈ L ∞ with µ(f ) = 0.
It is well-known for classical urn schemes with finitely many colors, that the fluctuations of the empirical distributions of colors in the urn depend crucially of whether the largest eigenvalue of the mean replacement matrix is larger or smaller than twice the real part of the second largest eigenvalue.This incites us to introduce In particular, ρ > 1/2 if and only if E s (C) > µ(a), which we interpret as reinforcement being stronger than innovation.
(ii) If ρ < 1/2, and further a slightly stronger version of (4), holds, then the sequence of processes converges in the sense of finite dimensional distributions to the Gaussian bridge G (br) .
(iii) If ρ = 1/2 and (13) holds, then the sequence converges in the sense of finite dimensional distributions to the Gaussian bridge G (br) .
Remark 3. (i) Theorem 2(ii-iii) should be viewed as weaker analogs of Donsker's limit theorem for empirical distributions.It would be interesting to strengthen convergence in the sense of finite dimensional distributions here to a uniform central limit type theorem on appropriate classes of functions as in Dudley [9].
(ii) For instance, Simon's model with words given by uniform random variables discussed at the beginning of this introduction corresponds to the case where S = [0, 1], and P s is the law of a pair (C, ξ) where C is a Bernoulli variable with parameter p and ξ = (1 − C)δ V , with V a uniform variable on [0, 1] independent of C. hence P s does not depend on s, we may take (iii) It would also be interesting to further investigate fluctuations around the second order limit in the super-critical case ρ > 1/2.In the setting of Simon's model, Bertenghi [7] observed that these are asymptotically normal; see also [17].
The number C of copies of the sampled ball which are returned at a typical step does not play much of a role in (9), meaning that for such urn schemes with innovation, reinforcement does not impact the first order asymptotic of the empirical measures of colors in the urn.The effects of the reinforcement on the fluctuations can now be analyzed by inspection of the formula (11) for the covariance.In short, consider first the case C ≡ 0 without reinforcement, where at each step one only returns in the urn the atoms of a point process ξ.By ( 9), the empirical distribution of colors still converges to µ, and by Theorem 2, the fluctuations are governed by a Gaussian bridge G (br) 0 with covariance The comparison with (11) shows that the reinforced scheme with C ≡ 0 induces an additional white noise component in the fluctuations, which is seen from the term µ(f g) E s (C 2 )µ(ds).Keep in mind that the reinforcement variable C also impacts the eigenvalues λ 1 and λ 2 , and in particular that increasing C while keeping the innovation ξ unchanged also increases the ratio ρ.
It has been briefly mentioned at the beginning of this introduction that urns schemes with infinite colors, or measure-valued Pólya processes, have been considered in several recent works [3,4,5,14,15,20,21], and we shall now recast our contribution from this perspective.In [5] and [20], the authors consider urns processes as Markov chains (U n ) n≥0 of random measures (not necessarily point processes) on a measurable space S, such that conditionally given U 0 , . . ., U n , a random color σ is sampled according to the probability measure proportional to U n and then U n+1 = U n + R σ , where (R s ) s∈S is a replacement kernel of positive measures on S. A more general version where now the replacement kernels are random has been developed then in [14] and [21], and the framework there encompasses ours.Convergence of empirical distribution of colors has been established in [21,Theorem 1] or [14,Section 6], or also [3], under certain additional conditions which however need not to be fulfilled in our setting.Fluctuations for measure-valued Pólya processes have been considered very recently in [15], where results bearing the same flavor as our Theorem 15 are established.However [15] imposes a strong balance condition that would imply in our case that the total number of returned ball at each step is some constant, independently of the sampled color.Thus, although general urn schemes with random replacement kernels encompass the setting of the present work, and Theorems 1, 2 and 15 are close relatives to results there, the present contribution cannot be reduced to these works either.We have chosen here to consider random replacement kernels induced by point processes only rather than more general random measures as in the works that were just mentioned above, mostly to keep the connection with Pólya's original model as close as possible.
The general route that we will follow to establish Theorems 1 and 2 is well paved since the work by Athreya and Karlin [1] (see also [13]) for urn schemes with finitely many colors, or more generally for finite-dimensional stochastic approximation [6].We shall exhibit natural martingales, investigate the asymptotic behavior of their quadratic variations and apply a version of the martingale central limit theorem.Here and there, we shall take a few shortcuts that where not fully available at the time when [1] was written, such as stable convergence (see [12, Section VIII.5c]).Another rather minor difference is that we do not use the embedding of urn schemes in a continuous time Markov branching process known as poissonization, but merely a Poisson subordination.More precisely, Theorem 2 will be deduced from Theorem 15, which is a more precise functional limit theorem for the continuous time version of the urn scheme with innovation, analogous to results of Janson [13] in the setting of urns with finitely many colors.
The plan for the rest of this text is as follows.Some further notation and preliminary observations on the replacement kernel and on Poisson subordination are presented in Section 2. The foundations of our analysis, namely the natural martingales, are laid down in Section 3; these yield Theorem 1.The large time behavior of quadratic variations is determined in Section 4. In Section 5, we establish a functional limit theorem (Theorem 15), and finally derive Theorem 2 from the latter.

The replacement kernel
We introduce first some notation, referring to [16,Chapter 1] for background on the notions we shall use.We consider some Borel space endowed with its Borel sigma-field (S, S), and write N S for the space of finite counting measures on S; N S is also a Borel space.Recall that the replacement kernel (P s ) s∈S defines for every s ∈ S a probability distribution on {−1, 0, 1, 2, . ..}×N S which determines the join law of the number of copies C and the point process of innovation ξ when the sampled ball has color s.We assume throughout the rest of this text and without further mention that the requirements (2), ( 3) and (4) are fulfilled.
For every s ∈ S, k ≥ −1 and Φ : N S → R + measurable, there is the identity In particular, the intensity measure of ξ under P s is given for every f ∈ L ∞ by Observe also from the notation (5) that and We formalize the urn scheme with innovation depicted in the introduction and construct a sequence of point processes (U n ) n≥0 on S as follows.We start from some arbitrary U 0 = 0 in N S .Next for every n ≥ 0, we sample a color σ n according to the current empirical distribution U n and independently of U 0 , . . ., U n−1 , that is Then given σ n = s, we set where (C n+1 , ξ n+1 ) is distributed according to P s and independent of the preceding steps.The point process U n gives the composition of the urn after n steps.
The sequence (U n ) n≥0 is a Markov chain on N S , with one-step transitions given by ( 16).

Poisson subordination
One of the most efficient and best know tools in the study of urn schemes is the so-called poissonization, which, roughly speaking, consists of embedding the urn scheme in a continuous time multitype branching process.See for instance, [1], [2, Section V.9], [19, Section 4.6] or [13].Here, we shall rather use subordination2 based on an independent Poisson process.The advantage is that this both circumvents the possibly delicate issue of depoissonization, and still makes some computations (notably for quadratic variations) much simpler than in discrete time.The drawback is that the independence between the counting processes of balls of different colors induced by poissonization is lost here; however this will not be an issue for us.Specifically, let N = (N (t)) t≥0 denote a standard Poisson process, so the Stieltjes measure dN (t) is a Poisson point process on [0, ∞) with intensity the Lebesgue measure.We assume that N is independent of the urn scheme and use the jump times of N as the set of times at which a ball is sampled from the urn scheme.Specifically, we set and think of X = (X t ) t≥0 as the process describing the composition of the urn as time passes.We write respectively P and E for the probability measure and the mathematical expectation induced by this process, and also (F t ) t≥0 for its natural filtration.Occasionally, we may prefer to write P m and E m to stress that the initial composition of the urn is U 0 = X 0 = m for some non-zero m ∈ N S .We shall need some elementary bounds on the growth of the number of balls in the urn.
Proof.(i) Observe from ( 4) and Markov's inequality that We infere that there is some integrable variable, whose expectation is denoted by γ, that dominates stochastically the number C + ξ(1) of balls which are added to the urn at any step.It follows from the law of large numbers that the total number of balls in the urn growth at most linearly, and we conclude applying the law of large numbers for the Poisson process N .
(ii) Indeed, the process (X t (1) − X 0 (1)) t≥0 which counts the number of added balls can be bounded from below by some Poisson process with rate inf s∈S P s (C + ξ(1) ≥ 1), which is strictly positive by (4).The assertion is plain from elementary large deviations estimates.
Subordination turns Markov chains into Markov jump processes.In our setting, we get from ( 16) that the infinitesimal generator G of X is given for any bounded measurable functional Φ : N S → R and any non-zero measure m ∈ N S by with the notation m(ds) = m(ds)/m (1).
The starting point of our analysis based on the following observation.We shall consider linear functionals on N S , that is of the type Φ(m) = m(f ) for some f ∈ L ∞ .Of course, such functionals are not bounded as soon as the support of f is infinite; nonetheless (17) still makes sense and we have for m = 0 where R : L ∞ → L ∞ is the average replacement operator defined by (6).
In the next section, we will use these observations to exhibit simple martingales.Recall from ( 7) and ( 8) that λ 1 and λ 2 appear in this setting as the two eigenvalues of R.

Key martingales
Recall that the function a ∈ L ∞ appearing in (3) modulates the intensity of innovations.We set , t ≥ 0.
We also introduce for every f ∈ L ∞ with µ(f ) = 0 Beware that the factor in the exponential is λ 1 in the first definition and λ 2 in the second.
Lemma 5.The processes M (a) and M (f ) are square integrable martingales.
Proof.Recall first from ( 4) and ( 16) that It follows that E(X t (1) 2 ) < ∞ for every t ≥ 0, and therefore processes M (a) and M (f ) are square integrable.We will henceforth focus on M (a), as the proof of the martingale property for M (f ) relies on similar arguments.Our main task is to check that the process X t (a) − λ since then it follows from the Itô formula that M (a) is a martingale too.In this direction, we consider the bounded functional on N S given by Φ b (m) = m(a)1 {|m(a)|≤b} for some b > 0. We know that is a martingale, and we analyze the limit as b → ∞.Plainly, Φ b (X t ) increases to X t (a) ∈ L 1 (P) as b increases to ∞, and therefore Φ b (X t ) converges to X t (a) in L 1 (P).Furthermore, for every m ∈ N S , and since, thanks to (4), we deduce from ( 17) by dominated convergence that for every r ≥ 0, there is the almost-sure convergence where for the second identity, we used ( 14) and (15).
On the other hand, the bounds above show that for any m ∈ N S , This enables us to apply dominated convergence, and for every t ≥ 0 This completes the proof of (18).
The martingale M (a) is positive, hence has a terminal value Lemma 6.The martingale M (a) is a bounded in L 2 (P) and M ∞ (a) > 0, P-a.s.
Proof.We will check that the terminal value of the oblique bracket M (a) ∞ is integrable.To start with, observe from the urn scheme that for every n ≥ 0, Recall the assumption (4) and write β < ∞ for the bound above.Since X(a) is obtained from U (a) by subordination via an independent Poisson process, the Stieltjes measure of the oblique bracket of the semimartingale X(a) satisfies d X(a) t ≤ βdt, and then, by stochastic calculus, dr.
We next introduce the time substitution and stress that T (t) < ∞ a.s.for any t > 0, thanks to Lemma 4(i).Write We shall establish below that for any initial value m ∈ N S , This entails the integrability of M (a) ∞ , hence the boundedness of M (a) in L 2 (P m ), and more precisely Note first that since T (t) is an F t -stopping time and M (a) a nonnegative martingale, E m (M T (t) (a)) ≤ m(a), and hence Next, consider the function f = a − µ(a)1, which has µ(f ) = 0. Again by optional sampling applied now to the martingale M (f ), we have for every bounded stopping time τ that (1) .
), and we get by dominated convergence that The proof of ( 20) is now complete, and we turn our attention to the strict positivity of the terminal value.An application of Chebychev's inequality in (21) yields .
Observe that for any t 0 > 0, there is the equivalence So by the Markov property at time t 0 , To conclude that P m (M ∞ (a) = 0) = 0, we only need to let t 0 → ∞ and note that lim t0→∞ X t0 (a) = ∞ a.s., since a is bounded away from 0 and the total number of balls in the urn grows to ∞ as time goes to ∞ a.s.
We next turn our attention to the predictable quadratic variation of the martingale M (f ).
where for every non-zero m ∈ N S , Proof.The urn scheme ( 16) yields, in the notation of the statement, that for any Since X is obtained from U by subordination via an independent Poisson process, its optional quadratic variation process The exponential term in the right-hand side is a continuous adapted process, so the predictable compensator see [12,Theorem I.3.17].This is the formula in the statement.
Lemma 7 enables us to bound the square moments of M (f ), and yields the following.
Proof.In the case f = a, the claim amounts to the convergence of the martingale M (a) to its terminal value; see Lemma 6.By linearity, it thus suffices to check that for every f ∈ L ∞ with µ(f ) = 0, one has = 0 a.s. and in L 2 (P).
In this direction, recall the time-substitution (19) and rewrite the above in the simpler form lim t→∞ e −(λ1−λ2)t M T (t) (f ) = 0 a.s. and in L 2 (P).
The assumption (4) ensures that and we deduce from Lemma 7 that the time-changed process (M T (t) (f )) t≥0 is still a square integrable martingale with oblique bracket The process e −(λ1−λ2)t M T (t) (f ) is a supermartingale, and it therefore suffices to verify that the limit in (22) takes place in L 2 (P).We compute We now see from (20) and the fact that For this, we apply Corollary 8 with f = 1 and get lim t→∞ This yields (24) by integration.
Obviously, Theorem 1 follows from Corollary 10 and the law of large numbers for the Poisson process N .

Asymptotic behaviors of quadratic variations
Limit theorems for martingales are of course the key ingredient for Theorem 2. Their applications require controlling quadratic variations, and for this we use mainly Lemmas 6 and 7 and (24).We start with a simple observation; recall the notation (12).Corollary 11.Suppose ρ > 1/2.Then for every f ∈ L ∞ with µ(f ) = 0, there is the convergence a.s. and Proof.Recall the notation (23) and apply Lemma 7 to get The martingale M (f ) thus converges a.s. and in L 2 (P).It only remains to use (24).
We next turn our attention to the case ρ ≤ 1/2 and first consider the growth of oblique brackets.Recall the notation (10).

Corollary 12.
The following assertions hold for every f ∈ L ∞ with µ(f ) = 0: (ii) If ρ = 1/2, then we have Proof.Recall the notation of Lemma 7 and observe from Corollary 10 and the requirement µ(f We deduce from Lemma 7 and ( 24) that almost surely By dominated convergence, this yields our claims.
We shall also need the following bounds on the p-variation of M (f ).
Lemma 13.Assume (13).If ρ < 1/2, then we have for any Proof.Note first that for any ε > 0 and t ≥ 0, there is the inequality Choosing ε sufficiently small and using Lemma 4 and Corollary 8 for f = 1, we get Next, computing predictable compensation in the same way as we did in the proof of Lemma 7, we get for any p > 0 Provided that pλ 2 ≤ 2λ 1 , that is pρ ≤ 2, we have by Jensen's inequality and (25) that The statement is now clear when ρ < 1/2, and in the case ρ = 1/2, we may just take any p ∈ (2, α]. The asymptotic behavior of optional quadratic variations can be deduced in turn from Corollary 12 and the following observation.Lemma 14. Assume (13) and that ρ ≤ 1/2.Then for every f ∈ L ∞ with µ(f ) = 0, we have Proof.Suppose first that ρ < 1/2.By Corollary 12, it suffices to verify that where D(f We can bound the quadratic variation of D for any 2 < p ≤ 4 by Choosing p as in Lemma 13, and applying the Jensen and the Burkholder-Davis-Gundy inequalities, we get so (26) is verified.
In the case ρ = 1/2, the same argument shows that the martingale D is bounded in L p/2 (P) and hence converges a.s.

Applying stable limit theorems
The purpose of this section is to establish the following functional limit theorem from which Theorem 2(ii-iii) will readily follow.The space D(R + , R) of càdlàg real-valued paths is endowed with the Skorokhod topology; see [12,Section VI.1].Recall the notation (10), and let also W = (W (t)) t≥0 denote a standard real Brownian motion.Theorem 15.Assume (13) and take any f ∈ L ∞ with µ(f ) = 0.
(i) If ρ < 1/2, then there is the convergence in distribution on D(R + , R) in the sense of Skorokhod Proof of Theorem 15.Again, we shall only prove (i) as the argument in the critical case ρ = 1/2 requires only simple modifications.The claim is plain from Corollary 12(iii) when σ 2 (f ) = 0, so we shall also assume henceforth σ 2 (f ) = 0.The proof relies on general results on the so-called stable convergence for martingales (see [12, Section VIII.5.c] for background) and the technical results we established so far.
We shall apply [12,Theorem VIII.5.50] for the martingale Y resulting from M (f ) after the deterministic time change t → t 1/(1−2ρ) , namely More precisely, the statement there requests, for the sake of simplicity, boundedness of the jumps of Y (see [12, Eq.VIII.5.48]); however its proof only uses the weaker condition VIII.3.14, which, in the present setting, follows from Lemma 13.Next, we have from Corollary 12(i) and Lemma 14, This is the condition VIII.5.49 in [12].Theorem VIII.5.50 and Proposition VIII.5.33(ii) from [12] now entail that the sequence of pairs η, (n −1/2 Y nt ) t≥0 converges in law to (η, ηW ), where W is a standard Brownian motion independent of η.Recall also from Lemma 6 that η > 0 a.s.It follows that Since N (t) ∼ t a.s., this yields n −ρ U n (f ) ∼ L ′ f and the claim is proved with L f = λ −1 L ′ f .(ii) Fix first f ∈ L ∞ with µ(f ) = 0. Write 0 < γ(1) < γ(2) < . . .for the increasing sequence of the jump times of the Poisson process N , so N (γ(n)) = n and U n (f ) = X γ(n) (f ).Since lim n→∞ n −1 γ(n) = 1 a.s., we deduce from Theorem 15(i) that there is the convergence where G is the centered Gaussian process on L ∞ with covariance given by (11) and G (br) its bridge version.It now follows from Theorem 1 and Slutsky's lemma that lim Since U n and G (br) are both translation invariant (in the sense that U n (f + c) = U n (f ) and G (br) (f +c) = G (br) (f ) for any c ∈ R), the convergence in distribution above holds for any fixed f ∈ L ∞ , without the restriction that µ(f ) = 0. Finally, that the latter holds more generally jointly for any finite family of functions in L ∞ stems from the Cramér-Wold device.
(iii) The argument is similar to (ii).

nt 0 dr X r ( 1 )
, t ≥ 0, converges in law to the time-changed Brownian motion W (t 1−2ρ ) t≥0 .To complete the proof, it suffices to recall from (24) that lim follows readily from Corollary 10 and Theorems 15 and 1.Proof of Theorem 2. (i) By linearity, it suffices to consider the case when µ(f ) = 0. We write firstn 1−ρ U n (f ) = n U n (1) × n −ρ U n (f ),and recall from Theorem 1 that the first term in the product in the right-hand side converges to 1/λ 1 as n → ∞.We next rewrite Corollary 10 in the formlim t→∞ N (t) t ρ N (t) −ρ U N (t) (f ) = L ′ f , a.s.