On The Traces Of Laguerre Processes

Almost sure and $L^k$-convergence of the traces of Laguerre processes to the family of dilations of the standard free Poisson distribution are established. We also prove that the fluctuations around the limiting process, converge weakly to a continuous centered Gaussian process. The almost sure convergence on compact time intervals of the largest and smallest eigenvalues processes is also established

The continuous n × n−matrix-valued process L m,n (t) = B * m,n (t)B m,n (t), t ≥ 0, is known as Laguerre process (or complex Wishart process) of size n, of dimension m and starting from L m,n (0) = B * m,n (0)B m,n (0).For n = 1, L m,1 is a squared Bessel process.Let Σ m,n (t) t≥0 = { 1 2n L m,n (t)} t≥0 be a Laguerre processes scaled by 1 2n and let be the n-dimensional stochastic process of eigenvalues of Σ m,n (t) t≥0 .
In the case of real Wishart processes (i.e.B m,n (t) has real entries) and m > n − 1, Bru [4] proved that if the eigenvalues start at different positions then they never meet at any time and furthermore they are governed by a diffusion process satisfying an Itô Stochastic Differential Equation (ISDE).
A special feature of this system of ISDEs is that it has non smooth drift coefficients and the eigenvalues processes do not collide.When m > n − 1, this SDE has a unique solution in the sense of probability law [7,Theorem 4].
Throughout the paper we shall assume that m > n − 1 and (1).We denote by Tr the usual unnormalized trace of a matrix and tr = 1 n Tr is the normalized one.Consider the empirical process where δ x is the unit mass at x.In the present paper we are concerned with functional limit theorems for the p-moment or p-trace processes associated with µ (m,n) t for any p ≥ 1.The dynamical behavior of the extreme eigenvalues processes is also investigated.Specifically, we consider propagation of chaos and asymptotic fluctuations for the one-dimensional processes M m,n,p (t) t≥0 : n ≥ 1 defined by the semimartingales An important role in this paper is played by the family of dilations of free Poisson law.Recall that the free Poisson distribution (or Marchenko-Pastur distribution) µ f p c , c > 0, is the probability measure on + defined by where It was shown by Marchenko and Pastur [23] that µ f p c is the asymptotic distribution, when t = 1, of the empirical spectral measure (3) when lim n→∞ m n = c.This explains why when c < 1 this distribution has an atom at zero with mass (1 − c), since in this case the Laguerre matrices are singular having zero eigenvalues.In this work we always consider the case c ≥ 1.
In this work we also consider the family µ c (t) t>0 of dilations of the free Poisson distribution given by µ with µ c (0) = δ 0 .For the p-moment we use the notation For a fixed t > 0, the study of different aspects of traces or moments of Wishart random matrices has been considered by Marchenko and Pastur [23], Oravecz and Petz [24], Voiculescu-Dykema and Nica [27], amongst others.An important role in those papers is played by the moments of the free Poisson law µ f p c .In the case of complex Hermitian Brownian motions B n,n (t); t ≥ 0 , n ≥ 1, the corresponding systems of eigenvalues are called Dyson Brownian motions.In the study of functional limit theorems for these processes (see for example [5], [25]), a key role is played by the family of semicircle laws which agrees with the dilations of the standard semicircle law, which is the spectral distribution of the free Brownian motion.However, in the case of Laguerre process, we show that the limit is the family of dilations of free Poisson laws which is different from the family of free Poisson laws, obtained by taking c t as the parameter.
The paper is organized as follows.For the sake of completeness, in Section 2 we recall a known result on the characterization of the families of dilations of free Poisson laws in terms of an initial valued problem for their Cauchy transforms.We also present a functional recursive equation for the families of moments of free Poisson laws.
In Section 3 we prove, for the Laguerre model, uniform almost surely and in L k laws of large numbers.We show that the sequence of measure-valued processes µ (m,n) t converges almost surely, as lim n→∞ m n = c, to the family µ c (t) t≥0 (which for t = 1 agrees with µ f p c ) in the space of continuous functions from + into probability measures in , endowed with the uniform convergence on compact intervals of (Theorems 3.1 and 3.3).We also address the question of the weak convergence of the fluctuations of the moment processes . It is shown that for each r ≥ 1, V m,n,r converges, as lim n→∞ m n = c, to a one-dimensional Gaussian process Z r given in terms of the previous (r − 1)th limiting processes Z 1 ,...,Z r−1 , and a Gaussian martingale (Theorem 3.7).By using an upper large deviations bound for the empirical measure valued process obtained in [5], we derive an upper estimate of the deviation of the moment processes from a given deterministic measure valued process ν in terms of the entropy of ν (Proposition 3.5).
In Section 4 we prove the almost sure convergence, as lim n→∞
The Cauchy transform for the free Poisson distribution is For the laws µ c (t) t≥0 , writing G c,t (z) for G µ c (t) (z), we have the relation The following characterization of the family of distributions µ c (t) t≥0 of dilations of free Poisson distribution in terms of an initial valued problem for the corresponding Cauchy transforms was proved in [5, Corollary 3.1].

Proposition 2.1
The family µ c (t) t≥0 is characterized by the property that its Cauchy transforms is the unique solution of the initial value problem Remark 2.2.a) From Lemma 3.3.9 in [17], a characterization of the family µ Poisson laws can be obtained in terms of an initial valued problem for the corresponding Cauchy transforms.
b) From [11, Remark 6.9] or [24, Proposition 11] we have the relation and hence The following functional recursive equation for the families of moments of free Poisson laws holds.
Proposition 2.3.For each r ≥ 1 and t > 0 Proof.The following formula for the moments of free Poisson distribution follows from the series expansion identifying the coefficients in the relation Now ( 14) is a consequence of ( 13) and (15).

Functional limit theorems for the trace processes
In this section we show the uniform a.s. and L k laws of large numbers for the moment pro- given by ( 4) and we prove the weak convergence of µ to a measure valued process satisfying an evolution equation.
Let Pr( ) be the space of probability measures on endowed with the topology of weak convergence and let C + , Pr( ) be the space of continuous functions from + into Pr( ), endowed with the topology of uniform convergence on compact intervals of + .As it is usual, for a probability measure µ and a µ-integrable function f we use the notation µ, f = f (x)µ(dx).

Propagation of chaos
The next goal is to prove uniform a.s. and L k , for each k ≥ 1, laws of large numbers for the trace processes M m,n,r .The first part of the next result gives useful recursive equations systems for the processes M m,n,r and for product of powers of them in terms of the martingales whose increasing processes are given by for any r ≥ 0.
In the following (m(n)) n is a sequence of positive integers with m(n For each k ≥ 0 and a multi-index where e 1 , ..., e k+1 is the canonical basis of k+1 and X m,n,r− 1 2 (t) is the martingale given by (16).
(ii) Assume that for each r, j ≥ 1, Then, for any k ≥ 0 and T > 0, there exists a positive constant K r k+1 , T such that In particular, for every r, j ≥ 1 and T > 0, there exist a positive constant K(r, j, T ) such that Moreover if (20) is satisfied, then Proof.(i) An application of Itô's formula to (2) gives that for The relation (18) follows from (26) with f (x) = x r . Define The Itô formula and (18) imply dr m,n i k+1 (ii) From ( 19) we obtain by induction over k the estimate and in particular sup Next, from Burkholder's inequality and ( 28) we obtain Now (21) (in particular ( 22)) follows easily from (19), by using ( 20), ( 28) and (29).
(iii) Now (29) implies On the other hand, using Chebyshev inequality and (29) we have that for each ǫ > 0 and thus sup 0≤t≤T The almost surely convergence in (24) follows from (31) and ( 18) by an induction argument, since the family µ c,r (t) t≥0,r≥1 satisfies uniquely the relation (14).

Remark 3.2. (a)
The recurrence relation (18) involves products of previous moment processes and therefore it cannot be used to estimate or to compute the expectation of the moments.For this reason we need to work with the products r m,n i k+1 (t) which are stable by an application of the Itô formula.
(b) Define the vector-valued process .
The proof of Theorem 3.1 implies the existence of a deterministic matrix D k such that The inductive relation (32) provides in particular the explicit computation of the moments M m,n,r (t); see [14] for real Wishart distributions, [13], [22] for complex Wishart distributions and [15] for the case of real Wishart processes.
Theorem 3.3.For any T > 0 and any bounded continuous function f : −→ we have That is, the empirical process µ (m(n),n) t t≥0 n≥1 converges a.s. in the space C + , Pr( ) .
Proof.From Theorem 3.1 follows the convergence (33) for continuous functions with compact support.
Now we prove the tightness of the sequence of processes µ in the space C + , Pr( ) .
From (26) and (29) it is easily seen that for every 0 which, by the well known standard criterion, shows that the sequence of continuous real processes is tight and consequently the sequence of processes µ is tight in the space C + , Pr( ) (see [10, pp. 107]).
Then, for any ǫ, T > 0 there exists a compact In particular we find a compact Γ ǫ,T ⊂Pr( ) such that and consequently, there exists a ǫ,T > 0 such that sup (in fact we have sup 0≤t≤T µ c (t) |x| ≥ a ǫ,T = 0 for a ǫ,T enough large).
From (34), ( 35) and an approximation argument we obtain (33) for all continuous and bounded functions.
Corollary 3.4.For any interval (a, b) ⊂ As a consequence of the above theorem we obtain Wachter's result on almost sure convergence of the empirical distribution of the eigenvalues to the measures µ f p c (see [11], [28]), since the moments determine uniquely the free Poisson law and also the well known result (taking t = 1) on the weak convergence of traces of complex Wishart matrices [11,Theorem 6.7].
In the case of Dyson Brownian motion the limit in the previous theorem is the family of semicircle laws which agrees with the dilations of the standard semicircle law (see [5], [16], [25]).However, in the case of Laguerre process, the limit is the family of dilations of free Poisson laws which is different from the family of free Poisson laws, obtained by taking c t as the parameter, which correspond to the free Poisson Lévy process.

Large deviations approach
The result of Theorem 3.3 is also a consequence of the following large deviations upper bound for the empirical process with good rate function (see [5, Theorem 3.2 and Corollary 3 The sequence µ satisfies in the space C [0, T ] , Pr( + ) the large deviation upper bound with good rate function S : for every closed subset F ⊂ C [0, T ] , Pr( + ) .
In particular the sequence µ is exponentially tight.

Fluctuations of moments processes
In this section we consider the asymptotic fluctuations of the moments processes M m(n),n,r (t) t≥0 around the corresponding moments µ c,r (t) t≥0 of the dilations of free Poisson distribution.Let V n,0 (t) = 0 and for r ≥ 1 From ( 26) we obtain that for f ∈ C 2 and t ≥ 0 The martingales play an important role (see [25] for the Dyson Brownian model).converges weakly in C( + , r ), when n goes to infinity, to a continuous centered Gaus- and quadratic variation Proof.Using the equality Burkholder's inequality and ( 22), we obtain for each 0 and thus the sequence Choose a subsequence N j := N n( j),k− 1 2 : 1 ≤ k ≤ r j≥1 which converges weakly to a limit N = .
By [18,Corollary 1.19,pp. 486] It follows that N is a continuous r-dimensional local martingale (in fact it is a martingale) and by [18,Theorem 6.1] we have that N j , N j , N j converges weakly to In particular N j , N j converges weakly to [N , N ] .From ( 13), ( 24) and (44) we deduce that .
By [18,Theorem 4.15] N is a process with independent increments and for s < t the random variable N t −N s is has centered Gaussian distribution.Therefore N is a continuous centered Gaussian martingale which satisfies (42) and (43).
In the final result of this section we show for each r ≥ 1, the fluctuation processes V n,r converge weakly to a one dimensional Gaussian process Z r , which is given by a recursive expression that involves Z 1 , ..., Z r−1 , the Gaussian martingale N r− 1 2 and the families of moments µ c,k (t) t≥0 , k = 1, ..., r − 1.
Theorem 3.7.For every k ≥ 1 assume that V n,k (0) converges weakly to V (0) k ∈ as n → ∞ and (20).Then, for each r ≥ 1, the r-dimensional process V n,k 1≤k≤r converges weakly in C( + , r ) to the unique in law continuous Gaussian process Z k 1≤k≤r given by where Z 0 = 0 and N k− 1 2 is given by (41).
Proof.From (45) and by induction on k it is easily seen that where P k (t) is a polynomial of order k, c, c j > 0 and j 1 , r 1 , ..., r j+1 ≥ 0.
It is clear that (46) implies that the distribution of the process Z k 1≤k≤r is uniquely determined by the distribution of the k-dimensional Gaussian martingale N k− 1 2 1≤k≤r . In particular Z k 1≤k≤r is a Gaussian process.
Taking f (x) = x k in (40) we obtain the equality By the Skorohod representation of the weak convergence (eventually in a new probability space) we can assume that If G is a nonnegative measurable functional defined on the space of continuous functions, then by the multivariate Cameron-Martin formula we have the equality for any 0 ≤ t ≤ T.
Proof.The fact that (57) (and (58)) is a martingale follows from the previous lemma and Novikov's criterion.
Next, if t 1 < t 2 from (2) and the fact that (57) is a martingale, we have

mn
= c, of the supremum over the interval [0, T ] of the largest eigenvalue process λ (m,n) n (t) t≥0 to 1 + c 2 T as well as the convergence for the infimum over the interval [0, T ] of the smallest eigenvalue process λ

Proposition 3 . 6 .
Next, for a r−dimensional martingale C = C j : 1 ≤ j ≤ r denote by [C, C] the r ⊗ r -valued process whose components are the quadratic variations C j , C k 1≤ j,k≤r .Assume (20) holds.Then, for each r ≥ 1 the r-dimensional martingale N n,k− 1 2 1≤k≤r