J Theor Probab DOI 10.1007/s10959-015-0627-7

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion Juan Carlos Pardo1 · José-Luis Pérez2 · Victor Pérez-Abreu1

Received: 30 September 2014 / Revised: 8 June 2015 © Springer Science+Business Media New York 2015

Abstract A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu (J Theor Probab 27:220–248, 2014). Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used. Keywords Matrix fractional Brownian motion · Measure-valued process · Free probability · Young integral · Fractional calculus Mathematics Subject Classification (2010) Secondary: 60H07

Primary: 60B20 · 46L54 · 60G22 ·

1 Introduction and Main Result Motivated by the fact that there is often a close correspondence between classical probability and free probability, Nourdin and Taqqu [12] recently introduced the non-

B

Victor Pérez-Abreu [email protected] Juan Carlos Pardo [email protected] José-Luis Pérez [email protected]

1

Department of Probability and Statistics, Centro de Investigación en Matemáticas, Apartado Postal 402, 36000 Guanajuato, GTO, Mexico

2

Department of Probability and Statistics, IIMAS-UNAM, Mexico City, Mexico

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commutative fractional Brownian motion (ncfBm). It appears as the limiting process in a central limit theorem for long-range dependence time series in free probability, in analogy to the classical probability case (see [18], for example). A ncfBm of Hurst parameter H ∈ (0, 1) is a centered semicircular process S H = (S H )t≥0 in a noncommutative probability space (A, ϕ) with covariance function  1   t 2H + s 2H − |t − s|2H . ϕ StH SsH = 2

(1.1)

The case S 1/2 is the well-known free Brownian motion introduced in [3]. The ncfBm is the only standardized semicircular process which is self-similar and has stationary increments. For the study of the ncfBm and the required free probability framework, we refer to Section 2 in [12] or Chapter 8 in [11]. In the present paper, we will deal mainly with the law (μtH )t≥0 of a ncfBm instead of the non-commutative process. Ever since the seminal paper by Voiculescu [19], it has been well known that free probability is a convenient framework for investigating the limits of the spectral distributions of random matrices (see for instance Section 5.4 in Anderson et al. [2]). On the functional asymptotic behavior side, Biane [3] proved that the free Brownian motion S 1/2 appears as the measure-valued process limit of n × n Hermitian matrix Brownian motions with size n going to infinity. Roughly speaking, this result gives a realization of the free Brownian motion S 1/2 as the spectral limit of well-known matrix-valued processes. On the other hand, for a fixed dimension n, the matrix-valued fractional Brownian motion was recently studied by Nualart and Pérez-Abreu [15]. It was shown that its corresponding eigenvalue process is non-colliding almost surely and a Skorohod stochastic differential equation governing this process was established. The main purpose of the present paper is to show that the ncfBm S H has a realization as the measure-valued process limit of n × n matrix fractional Brownian motions, as the size n goes to infinity. This gives a correspondence between classical fractional Brownian motion and non-commutative fractional Brownian motion. Our method uses the Skorohod and Young stochastic calculus for a multidimensional fractional Brownian motion as well as the fractional calculus. It is important to note that our methodology does not apply to the case H = 1/2 of the free Brownian motion. More precisely, let us consider a family of independent fractional Brownian motions starting from 0 with Hurst parameter H ∈ (1/2, 1), b = {{bi j (t), t ≥ 0}, 1 ≤ i ≤ j ≤ n} and define the symmetric matrix fractional Brownian √ motion of dimension n × n by B(t) by Bi j (t) = bi j (t) for i < j, and Bii (t) = 2bii (t). As we are interested in functional limit theorems for the eigenvalues of the fractional Brownian motion, for n ≥ 1 we will consider the following sequence of renormalized processes (B (n) )t≥0 , given by SH

1 B (n) (t) = √ B(t), for t > 0. n Following [15], it is possible to apply the chain rule to the Young integral to obtain the following equation for the eigenvalues of the process B (n)

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 (n) 1  t ∂Φi (n) λi (t) = √ (b(s)) ◦ dbkh (s), (n) n k≤h 0 ∂bkh

(1.2)

for any t > 0, i = 1, . . . , d, and where Φi(n) = λi(n) . Observe ∂Φi(n) (n) ∂bkh

(n) (n)

(n)

= 2u ik u i h 1{k=h} + (u ik )2 1{k=h}

(1.3)

(n) where u ik denotes the kth coordinate of the ith eigenvector of the matrix B (n) . The empirical measure-valued process which will be related to the functional limit theorems is n 1 (n) δλ(n) (t) , t ≥ 0, (1.4) μt = j n j=1

where δx denotes the unit mass at x. From the celebrated Wigner theorem in random (n) matrix theory, one has that for each fixed t > 0, μt converges a.s. to μsc t , the Wigner semicircle distribution of parameter t: μsc t (dx) =

1  4t − x 2 1−2√t,2√t  (x)dx, 2π t

see for instance [9,19,20]. The main result of this paper, stated in the framework of [5,17], is the following functional limit theorem for the empirical spectral measure-valued processes (n) {(μt )t≥0 : n ≥ 1} converging to the ncfBm. Let Pr(R) be the space of probability measures on R endowed with the topology of weak convergence and let C (R+ , Pr(R)) be the space of continuous functions from R+ into Pr(R), endowed with the topology of uniform convergence on compact intervals of R+ . (n)

Theorem 1 The family of measure-valued processes {(μt )t≥0 : n ≥ 1} converges weakly in C(R+ , Pr(R)) to the family (μt )t≥0 that corresponds to the law of a non-commutative fractional Brownian motion of Hurst parameter H ∈ (1/2, 1) and covariance (1.1). The case H = 1/2 of the free Brownian motion is known, see for instance [5,6,16, 17]. The proof of Theorem 1 is for H ∈ (1/2, 1), and it is done using results about the Young stochastic integral as well as fine estimations based on the fractional calculus. The rest of this paper is organized as follows. In Sect. 2 we derive the stochastic evolution of the empirical measure of the eigenvalues of the matrix fractional Brownian motion. In Sect. 3 we prove that the family {(μnt )t≥0 : n ≥ 1} is tight in C(R+ , Pr(R)). This is achieved by estimations of Young integrals by means of the fractional calculus. In Sect. 4, we show that the weak limit (μt )t≥0 , of the sequence of measure-valued (n) processes {(μt )t≥0 : n ≥ 1}, satisfies a measure-valued equation. In Sect. 5 we prove that the deterministic process (μt )t≥0 corresponds to the law of a non-commutative

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fractional Brownian motion. For this we show, using results in [19], that the process has semicircular finite-dimensional distributions, and covariance given by Eq. (1.1). For preliminaries on the stochastic calculus with respect to fractional Brownian motion, we refer to [10,11,13].

2 The Stochastic Evolution of the Empirical Measure of the Eigenvalues of a Matrix Fractional Brownian Motion As is usual, for a probability measure μ and a μ-integrable function f , we use the notation μ, f = f (x)μ(dx). Hence noting that the empirical measure (μt )t≥0 is a point measure, we have that for f ∈ Cb2 n

 1 (n) (n) μt , f = f (λi (t)). n

(2.1)

i=1

Therefore, applying the chain rule to the last equation, n  t



 1 (n) (n) (n) μt , f = μ0 , f + f (Φin (b(s))) ◦ dλi (s), t ≥ 0. n 0

(2.2)

i=1

(n)

In order to consider the evolution of the measure-valued process (μt )t≥0 , we prove the following result. (n)

Lemma 1 Let (μt )t≥0 be the empirical measure-valued process of the eigenvalues of the matrix fractional Brownian motion (B (n) (t))t≥0 . Then for each f ∈ Cb2 (R) and t ≥ 0 we have  n



 ∂Φi(n) 1  t n (n) (n) μt , f = μ0 , f + 3/2 f (Φi (b(s))) (n) (b(s))δbkh (s) n ∂bkh i=1 k≤h 0  t f (x) − f (y) 2H −1 (n) s μs (dx)μ(n) + H s (dy)ds x−y 0 R2 n  H  t

n + 2 f (Φi (b(s)))s 2H −1 ds. (2.3) n 0 i=1

Proof First we note that using (2.2) and (1.2) we obtain  n





∂Φi(n) 1  t n (n) (n) μt , f = μ0 , f + 3/2 f Φi (b(s)) (b(s)) ◦ dbkh (s). n 0 ∂b(n) i=1 k≤h

kh

Now we will be interested in replacing the Young integrals by Skorohod integrals in the above expression. To this end, we will prove that the condition of Proposition 3

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in [1] is satisfied. We will denote by D kh the Malliavin derivative with respect to bkh , for each 1 ≤ k ≤ h ≤ n. First note that  t 0

0

t

 Drkh

1 = 2H − 1 1 + 2H − 1



f 

 (n) ∂Φi n (Φi (b(s))) (n) (b(s)) |s ∂bkh t

0



− r |2H −2 dr ds

 2 n

∂Φi(n) f Φi (b(s)) (b(s)) s 2H −1 ds (n) ∂bkh



t

f



0

(n)

(Φin (b(s)))

∂ 2 Φi

(n)

∂(bkh )2

(b(s))s 2H −1 ds.

Therefore, using (1.3),  (n)   ∂Φi   ≤2 (b(s))  (n)  ∂bkh and so    (n) 2  t    ∂Φ 4 i

n 2H −1 

2H  f (Φ (b(s))) (b(s)) s ds i   ≤ 2H  f ∞ t < ∞. (n) ∂bkh  0  On the other hand, using (5.6) in [15], we obtain   (n)  t  ∂ 2 Φi 

n 2H −1  E  f (Φi (b(s))) (b(s))s ds  (n)  0  ∂(bkh )2     (n)  t  ∂ 2Φ   2H −1 i

≤  f ∞ E (b(s)) s ds  (n)  0  ∂(bkh )2   t  2 (n)   ∂ Φi  2H −1

=  f ∞ E  (b(s)) ds  s  ∂(b(n) )2  0 kh  t C1 H t < ∞. ≤ C1 s H −1 ds = H 0 Therefore, we can conclude that    t  2 Φ (n) ∂  i 2H −1  f (Φin (b(s))) (b(s))s ds   < ∞ P a.s. (n)  0  ∂(b )2 kh

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So, putting the pieces together, we obtain that  t 0

t 0

  (n) ∂Φi Drkh f (Φin (b(s))) (n) (b(s)) |s − r |2H −2 dr ds < ∞ P a.s. ∂bkh

Therefore, by Proposition 3 in [1] (see also Proposition 5.2.3 in [13]), we can express the Young integrals that appear in (1.2) in terms of Skorohod integrals. Therefore,  n (n)



 ∂Φi 1  t n (n) (n) μt , f = μ0 , f + 3/2 f (Φi (b(s))) (n) (b(s))δbkh (s) n 0 ∂b i=1 k≤h

kh  (n) ∂Φi kh

n + Dr f (Φi (b(s))) (n) (b(s)) |s−r |2H −2 dr ds n2 0 0 ∂bkh i=1 k≤h  n (n)

 ∂Φi 1  t n (n) = μ0 , f + 3/2 f (Φi (b(s))) (n) (b(s))δbkh (s) n ∂bkh i=1 k≤h 0  (n) 2  n ∂Φi H   t

n + 2 f (Φi (b(s))) (b(s)) s 2H −1 ds (n) n 0 ∂bkh i=1 k≤h  n 2 ∂ Φi(n) H  t n + 2 f (Φi (b(s))) (b(s))s 2H −1 ds. (n) 2 n ∂(bkh ) i=1 k≤h 0

  n H (2H −1)   t

t



On the other hand in p. 4280 of [15], we can find the following relation   ∂Φ (n) i (n)

k≤h

∂bkh

2 (b(s))

= 2.

(2.4)

Hence, using (2.4),  n (n)



 ∂Φi 1  t n (n) (n) μt , f = μ0 , f + 3/2 f (Φi (b(s))) (n) (b(s))δbkh (s) n 0 ∂b i=1 k≤h kh n  t  2H + 2 f

(Φin (b(s)))s 2H −1 ds n i=1 0  n 2H   t f (Φin (b(s))) 2H −1 s + 2 ds (n) (n) n i=1 j=i 0 λi (s) − λ j (s)  n (n)

 ∂Φi 1  t n (n) = μ0 , f + 3/2 f (Φi (b(s))) (n) (b(s))δbkh (s) n ∂bkh i=1 k≤h 0  t f (x) − f (y) 2H −1 (n) s μs (dx)μ(n) + H s (dy)ds 2 x − y 0 R

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J Theor Probab n  H  t

n + 2 f (Φi (b(s)))s 2H −1 ds. n 0 i=1

Here, in the third equality, we used the identity  ∂ 2 Φ (n) i

(n) 2 k≤h ∂(bkh )

(b(s)) = 2



1

(n) (n) j=i λi (s) − λ j (s)

. 

(see for instance, p. 4279 in [15]).

3 Tightness of the Family of Laws {µ(n) t : t ≥ 0} In this section we will prove that the family of the laws of the measured-valued (n) processes {(μt )t≥0 : n ≥ 1} is tight in the space C(R+ , Pr(R)). Proposition 1 The family of measures {(μ(n) t )t≥0 : n ≥ 1} is tight. Proof Using (2.1) it is easy to see that for every 0 ≤ t1 ≤ t2 ≤ T, n ≥ 1 and f ∈ Cb2 , n   



 1      (n)    (n) (n) (n)  μt2 , f − μt1 , f  ≤  f λi (t2 ) − f λi (t1 ) . n

(3.1)

i=1

We will assume that the eigenvalues are ordered in the following way (n)

(n)

λ1 (t) ≤ λ2 (t) ≤ · · · ≤ λ(n) n (t), for each t ≥ 0. Hence using Lemma 2.1.19 in [2] (the Hoffman–Wielandt inequality, see also [8]), and the fact that the eigenvalues do not collide for any t > 0 a.s., we deduce (n) (n) |λi (t2 ) − λi (t1 )|4

≤

 n 

2 (n) (n) λi (t2 ) − λi (t1 )

2

i=1

⎡

⎤  2 n   Bi j (t1 ) 2 Bi j (t2 ) 1 ⎦ , ≤⎣ − √ √ n n n i, j=1

for each i = 1, . . . , n. Therefore using the fact that the entries of the matrix fractional Brownian motion (B(t))t≥0 are independent, we obtain that there exists a constant C > 0 that does not depend on n such that   (n) (n) E |λi (t2 ) − λi (t1 )|4 ≤ C|t1 − t2 |4H ,

for all i = 1, . . . , n.

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Again, using that the function f is bounded and applying the Mean Value Theorem, we deduce (n)

(n)

(n)

(n)

| f (λi (r )) − f (λi (s))| ≤  f ∞ |λi (r ) − λi (s)|. Therefore using the above estimate in (3.1) and Jensen’s inequality, we obtain ⎡  ⎤ 

n   4 

 4   1   (n)   (n) (n) ≤  f 4∞ E ⎣ E  μ(n) λi (t2 ) − λi (t1 ) ⎦ t2 , f − μt1 , f  n i=1

≤ ≤

n  f 4∞ 

n

i=1

4 C f ∞ |t2

 4   (n)  (n) E λi (t2 ) − λi (t1 ) − t1 |4H .

Therefore, by a well-known criterion (see [7, Prop. 2.4]), we have that the sequence (n) of continuous real processes {(μt , f )t≥0 : n ≥ 1} is tight and consequently the (n) sequence of processes {(μt )t≥0 : n ≥ 1} is tight in the space C(R+ , Pr(R)). 

4 Weak Convergence of the Empirical Measure of Eigenvalues In the previous section, we proved that the family of measures {(μ(n) t )t≥0 : n ≥ 1} is tight in the space C(R+ , Pr(R)). Now we will proceed to identify the limit of any subsequence of the family. To this end we will first prove an estimate for the pth moment of the repulsion force between the eigenvalues of a matrix fractional Brownian motion, as the dimension goes to infinity. Lemma 2 For each p ∈ (1, 2), i = 1, . . . , n − 1, and for t ≥ 0 we have that   1 E = t − p H O(n p−2 ), as n → ∞, (n) (n) p |λi (t) − λi+1 (t)| uniformly with respect to t. n Proof For t ≥ 0, let us consider the eigenvalues {λ(n) (t)}i=1 of the matrix B (n) (t). Using (5.6) in [15] and (7.2.30) in [9], we have that the joint distribution of two consecutive eigenvalues is given by   (n) (n) P λi (1) ∈ dxi , λi+1 (1) ∈ dxi+1

= n

 √ √  (n − 2)! det K n1 xi n, xi+1 n dxi dxi+1 , n!

where  K n1 (u i , u i+1 ) =

123

Sn (u i , u i+1 ) + αn (u i ) Jn (u i , u i+1 )

Dn (u i , u i+1 ) Sn (u i+1 , u i ) + αn (u i+1 )



J Theor Probab

with Sn (u i , u i+1 ) =

n−1 

ϕ j (u i )ϕ(u i+1 ) +

 n 1/2

j=0

2

 ϕn−1 (u i )

R

δ(u i+1 − t)ϕn (t)dt,

∂ Dn (u i , u i+1 ) = − Sn (u i , u i+1 ), ∂u  i+1 In (u i , u i+1 ) = δ(u i − t)Sn (t, u i+1 )dt, R

Jn (u i , u i+1 ) = In (u i , u i+1 ) − δ(u i − u i+1 ) + β(u i ) − β(u i+1 ),  δ(u i − y)α(y)dy, β(u i ) = R  1 ϕ2m (u i )/ R ϕ2m (t)dt if n = 2m + 1 δ(u i ) = sign(u i ), αn (u i ) = 2 0 if n = 2m, and for each j ∈ N   √ −1/2 d j 2 exp(u i /2) − exp(−u i2 ). ϕ j (u i ) = (2 j! π ) du i j

(n)

(n)

By Proposition 5.4 in [15] we have that the process {(λ1 (t), . . . , λn (t))}t≥0 is H -self-similar, hence  E



1 (n)

(n)

|λi (t) − λi+1 (t)| p

 = t−pH E



1 (n)

(n)

|λi (1) − λi+1 (1)| p

.

Therefore, if we consider the following expectation, we have that  E



1 (n)

(n)

|λi (t) − λi+1 (t)| p

  1 (n − 2)! n! |x − xi+1 | p R R i √ √ × det[K n1 (xi n, xi+1 n)]dxi dxi+1 1− p n p   1 π − p H 2π =t n(n − 1) R R |u i − u i+1 | p 2n  √ √  × det K n1 (π u i / n, π u i+1 / n) du i du i+1 . = t−pH n

On the other hand, using (7.2.41) in [9] (see also Theorem 3.9.22 in [2]), we have that the joint density of the eigenvalues satisfies for any bounded interval I ⊂ R lim

n→∞

 √ √  π det K n1 π u i / n, π u i+1 / n = K (u i , u i+1 ) , 2n

(4.1)

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uniformly on u i , u i+1 ∈ I , where   K (u i , u i+1 ) = 1 − s 2 (r ) +



∞

s(t)dt

r

d s(r ) dr

 ,

with s(r ) = sin(πr )/πr and r = |u i − u i+1 |. Hence using the estimate (7.2.44) in [9], we note that  

1 K (u i , u i+1 ) du i du i+1 < ∞. p R R |u i − u i+1 |

Now we note that using (4.1) and Scheffe’s Theorem (see p. 215 in [4]) the following holds 

 √ √  1 π det K n1 (π u i / n, π u i+1 / n) du i du i+1 p n→∞ |u |+|u −u |>1 |u i − u i+1 | 2n i i+1  i 1 = K (u i , u i+1 ) du i du i+1 . |u − u i+1 | p i |u i |+|u i −u i+1 |>1 lim

Using that (4.1) holds uniformly on u i , u i+1 ∈ I , for any bounded interval I ⊂ R, then we can find N ∈ N, large enough, such that  π  √ √     det K n1 π u i / n, π u i+1 / n − K (u i , u i+1 ) < ε, for n ≥ N . 2n On the other hand, using polar coordinates, we obtain 

1 du i du i+1 p |u i |+|u i −u i+1 |≤1 |u i − u i+1 |  2π  √5 1 1− p ≤ r dθ dr | cos θ − sin θ | p 0 0   2π  2π 51− p/2 1 ≤ dθ + dθ 2− p | cos θ − sin θ |2 0 0      π  tπ 1 51− p/2 < ∞. 2π + tan + tan = 2− p 2 4 4

Therefore 

123

π  √ √  1   det K n1 (π u i / n, π u i+1 / n) p |u i |+|u i −u i+1 |≤1 |u i − u i+1 | 2n   1  , − K (u i , u i+1 )du i du i+1 ≤ ε p |u i |+|u i −u i+1 |≤1 |u i − u i+1 |

J Theor Probab

which in turn implies 

 √ √  1 π det K n1 π u i / n, π u i+1 / n du i du i+1 p n→∞ |u |+|u −u |≤1 |u i − u i+1 | 2n i i+1  i 1 = K (u i , u i+1 )du i du i+1 . p |u i |+|u i −u i+1 |≤1 |u i − u i+1 | lim

Hence we finally obtain  lim n

2− p

E



1 (n)

(n)

|λ (t) − λi+1 (t)| p  i 1 = πt−pH K (u i , u i+1 ) du i du i+1 . |u − u i+1 | p R R i

n→∞



The previous lemma will allow us to prove the following result related to the convergence of the multidimensional Skorohod integral that appears in (2.3), which in turn will enable us to identify the limit of any subsequence of the family of laws of processes {(μ(n) t )t≥0 : n ≥ 1}. Lemma 3 For any T > 0, any f : R → R such that f and f

are bounded, and p ∈ (1/H, 2), we have that  n  1    lim n→∞ n 3/2 

t

f



i=1 k≤h 0

  ∂Φi(n) (n) (Φi (b(s))) (n) (b(s))δbkh (s) ∂bkh

= 0, t ∈ [0, T ], (4.2)

in probability. Proof Let us use the following notation for the Skorohod integral with respect to the multidimensional fractional Brownian motion (b(t))t≥0 : 

t

g (b(s))δb(s) := i,n

0

 k≤h 0

t

i,n gkh (b(s))δbkh (s),

where (n)

i,n gkh (b(s)) := f (Φi (b(s)))

(n)

∂Φi

(n)

∂bkh

(b(s)), for each i = 1, . . . , n.

The following L p estimates will be very useful in what follows. For any p ≥ 1/H and i = 1, . . . , n,    

0

T

p    p p g i,n (b(s))δb(s) ≤ c p,T,H E(g i,n (b))L1/H ([0,T ]) +EDg i,n (b)L1/H ([0,T ]2 )

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J Theor Probab

 = c p,T,H  +E 0

T

E(g i,n (b(s))) p ds

0 T



T

1 H



pH

Ds g (b(r )) ds i,n

dr

(4.3)

0

where c p,T,H is a positive constant depending on p, H , and T . This last result is a consequence of Meyer’s inequalities: it appears for the onedimensional case in (5.40) of [13] and can be extended to the multidimensional case as in the proof of Proposition 3.5 of [14]. Now, we proceed to estimate each of the two integrals in the right-hand side of (4.3). Recalling the definition of g and using (2.4), it is clear, by Jensen’s inequality, that ⎡ 2 ⎤1/2   (n)  ∂Φ (n) i ⎦ (b(s)) E f (Φi (b(s))) (n) E(g i,n (b(s))) = ⎣ ∂b k≤h kh ⎡  (n) 2 ⎤1/2  ∂Φ i ≤  f ∞ ⎣ E (b(s)) ⎦ (n) ∂b k≤h kh = 21/2  f ∞ . Therefore, by Jensen’s inequality, we get 

T 0

  p E(g i,n (b(s))) p ds ≤ 2 p/2  f ∞ T.

For the second integral in the right-hand side of (4.3), we first compute an upper bound for the norm of the Malliavin derivative of g:  i,n Ds g i,n (b(r )) ≤ |(Dskh gkh (b(r )))| k≤h

⎛

2  (n) (n) | f

(Φi (b(r )))| ∂Φi (b(r )) = √ (n) n ∂bkh k≤h ⎞    (n) (n)  

2 | f (Φi (b(r )))|  ∂ Φi ⎟ + (b(r )) √ ⎠   2   n (n)   ∂bkh ⎫ ⎧  ⎪  ⎪ (n) ⎬  ⎨   ∂ 2Φ  i −1/2



≤ Cn (b(r ))  f ∞ +  f ∞  ,  2 ⎪  ⎪ (n) ⎩ k≤h  ∂bkh ⎭ 

for a positive constant C.

123

⎝

(4.4)

J Theor Probab

On the other hand, from p. 9 in [15] and Jensen’s inequality,       (n)      |u (n) (r )u (n) (r ) + u (n) (r )u (n) (r )|2    ∂ 2Φ ik jh jk ih  i 2    (b(r )) =   (n) (n)  (n) 2 λi (r ) − λ j (r )  k≤h  ∂b  k≤h  j=i kh

≤2

(n) (n) (n) 2   |u ik (r )u (n) j h (r ) + u jk (r )u i h (r )| k≤h i= j

≤2



(n)

(n)

|λi (r ) − λ j (r )|

−1 |λi(n) (r ) − λ(n) j (r )| ,

(4.5)

i= j

We have using (4.4) that  E



pH  1 Ds g i,n (b(r )) H ds dr 0 0 ⎫ p ⎞ ⎧  ⎛ ⎪  ⎪  T ⎬  ∂ 2 Φ (n) ⎨    ⎜ ⎟ i r p H  f

∞ +  f ∞ (b(r )) ≤ Cn − p/2 E ⎝  dr ⎠   2  ⎪  ⎪ (n) 0 ⎩ k≤h  ∂b ⎭ T

T

kh

Therefore, using (4.5), and Jensen’s inequality, we obtain, for p > 1, pH   T  T 1 i,n Ds g (b(r )) H ds dr E 0 0 ⎧ ⎛ ⎞p⎫  T ⎨ ⎬  (n) p p (n) ≤ K p n − p/2 r p H  f

∞ +  f ∞ E ⎝ |λi (r ) − λ j (r )|−1 ⎠ dr ⎩ ⎭ 0 i= j ⎧ ⎛ ⎞⎫  T ⎬ ⎨  (n) p p −p⎠ ≤ K p n − p/2 r p H  f

∞ +  f ∞ n p−1 E ⎝ |λi (r ) − λ(n) (r )| dr, j ⎭ ⎩ 0 i= j

(4.6) where K p is a positive constant depending on p. Now using Lemma 2, we can conclude that there exists a constant C( p) such that for large enough n ≥ 1,    (n) (n) E |λi (r ) − λ j (r )|− p ≤ C( p)r − p H n p−1 . i= j

So using the above estimate in (4.6), it is clear that there exist two constants K 1 (T, p, H ) and K 2 (T, p, H ) that depend on p, H and T such that  E 0

T



T

1 H

Ds g (b(r )) ds i,n

pH

 dr

0

≤ K 1 (T, p, H )n − p/2 + K 2 (T, p, H )n 3 p/2−2 .

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Therefore, putting all the pieces together, we have p   t   E  g i,n (b(s))δb(s) 0

≤ K 3 (T, p, H ) + K 1 (T, p, H )n − p/2 + K 2 (T, p, H )n 3 p/2−2 , where K 3 (T, p, f ) is a positive constant that depends on T, H , and p. In order to complete the proof, using Jensen’s inequality and the fact that p > 1, we observe  p n  t  1   i,n E  3/2 g (b(s))δb(s) n i=1 0   t p n    p−1 −3 p/2 i,n n E sup  g (b(s))δb(s) ≤n i=1

0≤t≤T

0

= n − p/2 (K 3 (T, p, H ) + K 1 (T, p, H )n − p/2 + K 2 (T, p, H )n 3 p/2−2 ). Hence for ε, T > 0, and taking p ∈ (1/H, 2), we obtain that    n  t   1  i,n   lim P  3/2 g (b(s))δb(s) > ε n→∞ n i=1

0

 1 ≤ lim p n − p/2 K 3 (T, p, H ) + K 1 (T, p, H )n − p/2 n→∞ ε  + K 2 (T, p, H )n 3 p/2−2 = 0. Therefore,   n  t  1   i,n   → 0, g (b(s))δb(s)  n 3/2  i=1

in probability as n goes to +∞.

0



With the previous results, we are ready to identify the weak limit of the sequence of (n) the measure-valued processes {(μt )t≥0 : n ≥ 1} as the solution to a measure valued equation. Theorem 2 The family of measure-valued processes {(μ(n) t )t≥0 : n ≥ 1} converges weakly in C(R+ , Pr(R)) to the unique continuous probability-measure valued function satisfying, for each t ≥ 0 f ∈ Cb2 (R),

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t

μt , f = μ0 , f + H

ds 0

f (x) − f (y) 2H −1 μs (dx)μs (dy). s x−y R2

(4.7)

(n)

Proof From Proposition 1, we know that the family {(μt )t≥0 : n ≥ 1} is relatively (n ) compact. Let us now take a subsequence {(μt k )t≥0 : k ≥ 1} and assume that it converges weakly to (μt )t≥0 . Therefore, by (2.3),  t



 (n ) (n ) μt k , f − μ0 k , f − H = +

nk   1  3/2

nk H n 2k

t

i=1 k≤h 0 n k  t 



i=1

0

f (x) − f (y) 2H −1 (n k ) k) s μs (dx)μ(n s (dy)ds x−y 0 R2 (n ) ∂Φi k (n ) f ( i k (b(s))) (n (b(s))δbkh (s) ) ∂bkhk (n k )

f f ( i

(b(s)))s 2H −1 ds.

(4.8)

Note that for 0 ≤ t ≤ T , the following limit holds P a.s.,   nk  t H   1 2H

  (n ) f

( i k (b(s)))s 2H −1 ds  ≤ T  f ∞ → 0, as k → ∞.  2  nk  2n k i=1 0 (4.9) Hence, using (4.9) and Lemma 3, it is clear that   nk   t (n )  1   ∂Φi k (n k )

 lim  3/2 f ( i (b(s))) (n ) (b(s))δbkh (s) k k→∞ n ∂bkh i=1 k≤h 0 k   nk  t H    (n k )

2H −1  + 2 f ( i (b(s)))s ds  = 0,  nk  i=1 0 in probability, and therefore there exists a subsequence (that without loss generality we will denote by (n k )k≥0 ) such that the same limit holds P a.s. Therefore, using (4.8)  t f (x) − f (y) 2H −1 μt , f − μ0 , f − H s μs (dx)μs (dy)ds 2 x−y

 0 R  k) k) , f − μ(n = lim μ(n t 0 , f k→∞  t f (x) − f (y) 2H −1 (n k ) s −H μs (dx)μs(n k ) (dy)ds = 0. 2 x − y 0 R k) Then we can conclude that any weak limit (μt )t≥0 of a subsequence (μ(n )t≥0 should t satisfy (4.7). Applying (4.7) to the deterministic sequence of functions

f j (x) =

1 , z j ∈ (Q × Q) ∩ C+ , x − zj

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and using a continuity argument, we get that the Cauchy–Stieltjes transform (G t )t≥0 of (μt )t≥0 satisfies the integral equation 

μ0 (dz) +H G t (z) = − R x−z





μs (dx)μs (dy) , t ≥ 0, z ∈ C+ . (x − z)(y − z)2 0 (4.10) From (4.10) it is easily seen that G t is the unique solution to the initial value problem 

∂ ∂t G t (z)

t

s

2H −1

ds

R2

∂ = H s 2H −1 G t (z) ∂z G t (z), μ0 (dx) G 0 (z) = − R x−z ,

t > 0, z ∈ C+ .

(4.11)

Therefore all limits of subsequences of (μ(n) t )t≥0 coincide with the family (μt )t≥0 , with Cauchy–Stieltjes transform given as the solution to (4.11), and thus the sequence 

{(μ(n) t )t≥0 : n ≥ 1} converges weakly to (μt )t≥0 .

5 Convergence to a Non-commutative Fractional Brownian Motion In this section, we prove that the deterministic process (μt )t≥0 corresponds to the law of a non-commutative fractional Brownian motion. The intuitive idea is as follows: by the (n) tightness of the sequence of processes {(μt )t≥0 ; n ≥ 1} in the space C(R+ , Pr(R)), (n ) the weak limit (μt )t≥0 of any subsequence {(μt k )t≥0 ; k ≥ 1} should satisfy (4.7) 2 for any t ≥ 0 and f ∈ Cb . Therefore by the uniqueness of solutions to Eq. (4.10), it is easy to check that G t (z) =

 1  2 z − 4t 2H − z , t > 0, z ∈ C+ , 2H 2t

which is the Cauchy–Stieltjes transform of a semicircle law with variance at time t > 0 given by t 2H , and hence the law of a non-commutative fractional Brownian motion at time t. Proof of Theorem 1 Let us recall that the sequence of processes {(μ(n) t )t≥0 : n ≥ 1} is tight in C(R+ , Pr(R)). This implies that the sequence is relatively compact, in other k) )t≥0 : k ≥ 1} that converges weakly to a words, there exists a subsequence {(μ(n t process that we denote by (μt )t≥0 in C(R+ , Pr(R)). Given the fact that the weak convergence of processes in C(R+ , Pr(R)) implies the convergence of the finite-dimensional distributions, then for each bounded and continuous function g: Rm → R, and for each sequence of times 0 ≤ t1 < · · · < tm , it follows that

 L + , (n k ) μt1 ,...,t , g → μt1 ,...,tm , g , as k → ∞. (5.1) m Let us now consider (B (n k ) (t))t≥0 the symmetric fractional Brownian matrix, such k) that the empirical measure of its eigenvalues [see (1.4)] is given by (μ(n )t≥0 . t First we will prove that the deterministic process (μt )t≥0 corresponds to the law of a semicircle process. To this end consider a set of points in time 0 ≤ t1 < · · · < tm

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and λ1 , . . . , λm ∈ R. Then for any polynomial Q, we have   m    m    1 (n k ) tr Q λi B (ti ) = Q λi xi μnt1k,...,tn (dx1 , . . . , dxm ) . E nk Rm 

i=1

i=1

Therefore 

  m   1 (n k ) tr Q λi B (ti ) lim E k→∞ nk i=1  m    = Q λi xi μt1 ,t2 ,...,tm (dx1 , . . . , dxm ) . Rm

(5.2)

i=1

From Theorem 2.2 in [19], we know that the random matrix X (n k ) = λ1 B (n k ) (t1 ) + · · · + λm B (n k ) (tm ), has a limit distribution, μ, ˜ which is a semicircle law. Hence using (5.2) we obtain that   Q(x)μ(dx) ˜ = Q (λ1 x1 + · · · + λm xm ) μt1 ,t2 ,...,tm (dx1 , dx2 , . . . , dxm ) . R

Rm

So if we define the function h: Rm → R by h(x1 , . . . , xm ) =

m 

λi xi .

i=1

Then the distribution μt1 ,...,tn ◦h −1 has a semicircle law. Therefore the process (μt )t≥0 is the law of a semicircular process. Now we proceed to identify the limit as the law of a non-commutative fractional Brownian motion. So first we will prove that (μt )t≥0 corresponds to the law of a centered semicircular process. To this end for t ≥ 0, we obtain, using (4.7) (with f (x) = x) the following  (5.3) xμt (dx) = x, δ0 = 0. R

Therefore, (μt )t≥0 is the law of a centered semicircular process. Finally in order to conclude the proof we compute the covariance: for t ≥ s ≥ 0, we obtain   (x y)μs,t (dx, dy) = lim (x y)μns,tk (dx, dy) 2 2 k→∞ R R   1  (n k ) tr B (t)B (n k ) (s) = lim E k→∞ nk  1  2H t + s 2H − |t − s|2H . = 2

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Noting that this holds for any subsequence, we can conclude that the whole sequence (n) {(μt )t≥0 : n ≥ 1} converges in law to the deterministic process (μt )t≥0 which is characterized by being the law of a non-commutative fractional Brownian motion. 

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