CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS*

Transcription

1 Communications on Stochastic Analysis Vol. 3, No. 1 (9) 1-14 Serials Publications CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS* JOHANNA GARZÓN Abstract. We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, our approximation does not make use of a kernel. 1. Introduction Fractional Brownian sheets have been studied by several authors for their mathematical interest and their applications. One of the first papers on the subject is [1]. Some types of approximations of fractional Brownian sheets have been obtained recently (e.g., [], [3], [9], [13], [14], [15]). In this paper we give a new type of approximation for certain values of the parameters by means of occupation time fluctuations of a stochastic particle model. The limits that are obtained in this way are a more general class of Gaussian random fields. We consider centered Gaussian random fields W (W s,t ) s,t with parameters (a i, b i ), i 1,, whose covariance is given by K W ((s, t), (s, t )) E (W s,t W s,t ) C(1) (s, s )C () (t, t ), (1.1) where each C (i) is of the form C (i) (u, v) u v with the following ranges for the parameters: r a i [(u r) b i + (v r) b i ]dr, i 1,, (1.) a i > 1, 1 < b i 1, b i 1 + a i. (1.3) C (i) is the covariance of weighted fractional Brownian motion with parameters (a i, b i ). Weighted fractional Brownian motions were introduced in [7]. We call W a weighted fractional Brownian sheet with parameters (a i, b i ), i 1,. In the case a 1 a (the weight functions are 1) W is a fractional Brownian sheet with Mathematics Subject Classification. Primary 6G6; Secondary 6G15, 6F5. Key words and phrases. Fractional Brownian sheet, weighted fractional Brownian sheet, approximation in law, long-range dependence. * Research partially supported by CONACYT grant F. 1

2 JOHANNA GARZÓN parameters ( 1 (1 + b 1), 1 (1 + b )). The case a 1 a b 1 b corresponds to the ordinary Brownian sheet. If b 1 b, and at least one a i is not, then W is a time-inhomogeneous Brownian sheet. Due to the covariance structure (1.1), (1.), many properties of W are consequences of those of weighted fractional Brownian motion. We will prove an approximation in law of W for a i and b i of the form a i γ i /α i, b i 1 1/α i, with γ i < 1 and 1 < α i ; hence the approximation is restricted to values of a i and b i such that 1 < a i and < b i < 1 + a i, i 1,. The approximations of fractional Brownian sheets in [], [15] are based on a Poisson random measure on R + R + and a kernel which appears in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion. The approximation in [3], analogous to the functional invariance theorem, also uses the kernel. Our approach does not use a kernel. We also use a Poisson random measure, but on R R instead of R + R + and in a different way from [], [15]. Some of the other approximations cited above are motived by simulation of fractional Brownian sheets. Our approximation is not intended for simulation, but rather to show that weighted fractional Brownian sheets emerge in a natural way from a simple particle model. In section we give the properties of W, in particular long-range dependence. In section 3 we describe the particle system and we prove convergence to W of rescaled ocupation time fluctuations of the system for the above mentioned values of the parameters.. Properties We consider R + with the following partial order: for z (s, t) and z (s, t ), z z iff s s and t t, z z iff s < s and t < t, and if z z we denote by (z, z ] the rectangle (s, s ] (t, t ]. We refer to elements of R + as times for simplicity of exposition. If X (X z ) z R is a two-time stochastic process, the increment of X over the + rectangle (z, z ] with z (s, t), z (s, t ) is defined by X((z, z ]) s,t X(s, t ) : X (s,t ) X (s,t ) X (s,t) + X (s,t). We denote the covariance of the increments of the process X over the rectangles ((s, t), (s, t )], ((p, r), (p, r )] by K X ((s, t), (s, t ); (p, r), (p, r )) Cov ( s,t X(s, t ), p,r X(p, r )). The covariance of W over rectangles is given by K W ((s, t), (s, t ); (p, r), (p, r )) (C (1) (s, p ) C (1) (s, p ) C (1) (s, p) + C (1) (s, p)) (C () (t, r ) C () (t, r ) C () (t, r) + C () (t, r)) Cov(Y (1) s Y (1) s, Y (1) p Y (1) p )Cov(Y () t Y () t, Y () r Y () r ), (.1) where Y (i) is weighted fractional Brownian motion with parameters (a i, b i ), i 1,.

3 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 3 The next theorem contains some properties of weighted fractional Brownian sheets. Theorem.1. The weighted fractional Brownian sheet W with parameters (a i, b i ), i 1,, has the following properties: (1) Self-similarity: (W hs,kt ) s,t d h (1+a 1 +b 1 )/ k (1+a +b )/ (W s,t ) s,t for each h, k >, (.) where d denotes equality in distribution. () W has stationary increments only in the case a 1 a. (3) Covariance of increments: For (, ) (s, t) (s, t ) (p, r) (p, r ), K W ((s, t), (s, t ); (p, r), (p, r )) s s hence u a 1 [(p u) b 1 + (p u) b 1 ]du t t v a [(r v) b + (r v) b ]dv, (.3) > if b 1 b >, K W ((s, t), (s, t ); (p, r), (p, r )) if b 1 b, < if b 1 b <. (4) The one-time processes (W s,t ) s (t fixed) and (W s,t ) t (s fixed) are weighted fractional Brownian motions (multiplied by constants) with parameters (a 1, b 1 ) and (a, b ), respectively. (5) s t E(( s,t W s,t ) ) 4 u a 1 (s u) b 1 du v a (t v) b dv. (.4) s t (6) lim ε,δ ε b1 1 δ b 1 E(( s,t W s+ε,t+δ ) 4 ) (1 + b 1 )(1 + b ) sa1 t a, (.5) lim T,S S (1+a1+b1) T (1+a+b) E(( s,t W s+s,t+t ) ) 4 1 u a 1 (1 u) b 1 du 1 v a (1 v) b dv, (.6) hence W has asymptotically stationary increments for long increments in R +, but not for short ones (if a 1, a ). (7) The finite-dimensional distributions of the process (S a 1/ T a / S,T W s+s,t+t ) s,t converge as T, S to those of fractional Brownian sheet with parameters ( 1 (1 + b 1), 1 (1 + b )) multiplied by /[(1 + b 1 )(1 + b )] 1/.

4 4 JOHANNA GARZÓN (8) Long-range dependence: for (s, t) (s, t ), (p, u) (p, u ) lim τ 1 b 1 κ 1 b K W ((s, t), (s, t ); (p + τ, u + κ), (p + τ, u + κ)) τ,κ b 1 b (1 + a 1 )(1 + a ) (p p)((s ) 1+a1 s 1+a1 )(u u)((t ) 1+a t 1+a ). (.7) (9) For θ >, we define the one-time process (Z t ) t (W t,θt ) t, i.e., the sheet restricted to a ray through the origen. (Note that Z is not a weighted fractional Brownian motion.) Then for b i < 1, i 1,, and not both b 1, b equal to, this process has the long-range dependence property lim τ 1 (b1+b) Cov(Z v Z u, Z t+τ Z s+τ ) τ θ1+a+b (b 1 + b ) (1 + a 1 )(1 + a ) (v+a1+a u +a1+a )(t s), u < v, s < t. (.8) Proof. Except for part (9), the proofs follow directly from the form of K W given by (1.1), (1.) and properties of weighted fractional Brownian motion [7]. We give an outline of the proof of part (9). We have, for u < v, s < t, Cov(Z v Z u, Z t+τ Z s+τ ) θ 1+a+b [C (1) (v, t + τ)c () (v, t + τ) C (1) (v, s + τ)c () (v, s + τ) C (1) (u, t + τ)c () (u, t + τ) + C () (u, s + τ)c () (u, s + τ)], (.9) C (1) (v, t + τ)c () (v, t + τ) C (1) (v, s + τ)c () (v, s + τ) [C (1) (v, t + τ) C (1) (v, s + τ)]c () (v, t + τ) + [C () (v, t + τ) C () (v, s + τ)]c (1) (v, s + τ) v v + r a 1 [(t r + τ) b 1 (s r + τ) b 1 ]dr v r a [(t r + τ) b (s r + τ) b ]dr v r a [(t r + τ) b + (v r) b ]dr r a 1 [(s r + τ) b 1 + (v r) b 1 ]dr, (.1) and similarly for the last two terms. The result follows from (.9), (.1) and the limits lim τ 1 b [(t + τ) b (t 1 + τ) b ] b(t t 1 ) τ and v lim τ b r a [(t + τ) b + (v r) b ]dr v1+a τ 1 + a. Remark.. There are three different long-range dependence regimes in property (9), and they are independent of a 1, a. The covariance of increments of Z has a power decay for b 1 + b < 1, a power growth for b 1 + b > 1, and a non-trivial limit for b 1 + b 1. We do not know if this property has been noted before for fractional Brownian sheets. It is worthwhile to observe that the non-gaussian

5 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 5 process (Y (1) t Y () θt ) t, where Y (i) are independent weighted fractional Brownian motions with parameters (a i, b i ), i 1,, has the same long-range dependence behavior. In [7] it is shown that A s,t t s ua (t u) b du, s < t, has the following bounds: If a, s, t T for any T > and constant M M(T ), and also if a <, s, t ε for any ε > and constant M M(ε), If a <, 1 + a + b >, s, t, A s,t M t s 1+b. A s,t M t s 1+a+b. Then it follows from (.4) that for < ε s < s < T, < ε t < t < T and i 1,, where δ i { E(( s,t W s,t ) ) M (s s) δ 1 (t t) δ, (.11) 1 + a i + b i if a i < and 1 + a i + b i >, 1 + b i otherwise. The next lemma allows us to prove the continuity of W. (.1) Lemma.3. [1], [1] Let X (X s,t ) s,t be a two-time stochastic process on a probability space (Ω, F, P ) which is null almost surely on the axes and such that there exist p >, a, b (1/p, ), such that (E( s,t X s+h,t+k p )) 1/p M h a k b. Then X has a modification X with continuous trajectories. Also, the trajectories of X are Hölder with exponents (a, b ), for a (, a 1/p), b (, b 1/p), that is, for any ω Ω exists M ω > such that for any s, s, t, t, s,t Xs,t (ω) M ω(s s) a (t t) b, s < s, t < t. Proposition.4. The weighted fractional Brownian sheet (W s,t ) s,t has a modification ( W s,t ) s,t with continuous trajectories. Also, the trajectories of W are Hölder with exponents (x, y) for any x (, 1 δ 1), y (, 1 δ ), where δ i are as in (.1). Proof. From the moments of the normal distribution and equations (.4) and (.11) we have (E( s,t W s+h,t+k r )) 1/r C ( s+h s Mh δ1/ k δ/, u a 1 (s + h u) b 1 du t+k t ) 1/ v a (t + k v) b dv with some constants C and M. Taking r > max {/δ 1, /δ } we have the conditions of Lemma.3, and the result follows.

6 6 JOHANNA GARZÓN 3. Approximation The random field W, for some values of the parameters a i, b i, arises as a limit in distribution of occupation time fluctuations of a system of particles of two types that move as pairs in R R according to independent stable Lévy processes. The system is described as follows. Given a Poisson random measure on R R with intensity measure µ, N, Pois(µ), from each point (x 1, x ) of N, come out two independent Lévy processes, from x 1 comes out ξ x 1, symmetric α 1 -stable, and from x comes out ζ x, symmetric α -stable ( < α i, i 1, ). Let N (N u,v ) u,v denote random measure process on R R such that N u,v represents the configuration of particles at time (u, v), N u,v (x 1,x ) N, δ (ξ x1 u,ζ x v ) (x 1,x ) N, δ ξ x1 For ϕ, ψ L 1 (R) (ϕ, ψ ) fixed, we write N u,v, ϕ ψ δ x1 δ ξ u ζ x, ϕ ψ v (x 1,x ) N, We define the occupation time process of N by L s,t, ϕ ψ s t δ u ζ x. (3.1) v (x 1,x ) N, ϕ(ξ x1 u )ψ(ζ x v ). (3.) N u,v, ϕ ψ dvdu, s, t, (3.3) and the rescaled occupation time fluctuation process by X T (s, t) 1 ( L T s,t t, ϕ ψ E( L T s,t t, ϕ ψ )), F T s, t, (3.4) where T is the time scaling and F T is a norming. We choose the intensity measure µ for the Poisson initial particle configuration as with µ(dx 1, dx ) µ 1 µ (dx 1, dx ) µ 1 (dx 1 )µ (dx ), µ i (dx i ) dx i / x i γi, γ i < 1, i 1,. (3.5) The homogeneous case corresponds to γ 1 γ and it gives rise to the usual fractional Brownian sheet. We will show that for F T F (1) T F () T with F (i) T T 1 (1+γ i)/α i, γ i < 1 < α i, i 1,, (3.6) the finite-dimensional distributions of the process X T converge in law as T to those of weighted fractional Brownian sheet with parameters a i γ i /α i, b i 1 1/α i, i 1,. In the case a 1 a we will also prove tightness. Theorem 3.1. If X T is the process defined in (3.4), γ i < 1 < α i, i 1,, with F T defined by (3.6), then the finite-dimensional distributions of X T converge as T to the finite-dimensional distributions of DW, where W is weighted fractional Brownian sheet with parameters a 1 γ 1 /α 1, b 1 1 1/α 1, a γ /α, b 1 1/α, and D is the constant ( ( 1 D ϕ(x)dx ψ(x)dx p α i 1 1 1/α p α i 1 (x) 1/ dx)) i x γ, i (3.7) R R i1 R

7 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 7 where p α t (x) is the density of the symmetric α-stable Lévy process, which is given by p α t (x) 1 exp { (ixy + t y α )} dy. π R Proof. For each k N, d 1,, d k R and (s 1, t 1 ),, (s k, t k ) R +, we must show that k k d j Xs T j,t j converges in law to d j W sj,t j as T, j1 which we do by proving convergence of the corresponding characteristic functions. From the fact that N, Pois(µ 1 µ ), we have for each θ R, { k } C T (θ) : E exp iθ d j Xs T j,t j { exp iθ F T j1 k } d j E( L T s j,t j, ϕ ψ ) j1 { exp R R [ 1 E (x1,x ) ( { iθ exp F T k j1 j1 })] } d j L T s j,t j, ϕ ψ µ 1 (dx 1 )µ (dx ), (3.8) where E (x1,x ) denotes expectation starting with one pair of initial particles in (x 1, x ), (see e.g., [11], mixed Poisson process). We also need the mean and the covariance of N. From the Poisson initial condition, the first and second moments are given by E ( N u,v, ϕ ψ ) E (x1,x ) ( N u,v, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R E (ϕ(ξu x1 )ψ(ζv x )) µ 1 (dx 1 )µ (dx ) (3.9) and R R E ( N u1,v 1, ϕ ψ N u,v, ϕ ψ ) E (x1,x ) ( N u1,v 1, ϕ ψ N u,v, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R + E (x1,x ) ( N u1,v 1, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R E (x1,x ) ( N u,v, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R E ( ϕ(ξ x 1 u 1 )ψ(ζ x v 1 )ϕ(ξ x 1 u )ψ(ζ x v ) ) µ 1 (dx 1 )µ (dx ) R R + E ( ϕ(ξu x1 1 )ψ(ζv x 1 ) ) µ 1 (dx 1 )µ (dx ) E ( ϕ(ξu x1 )ψ(ζv x ) ) µ 1 (dx 1 )µ (dx ), R R R R

8 8 JOHANNA GARZÓN hence, by the independence of ξ and ζ, and the Markov property, Cov ( N u1,v 1, ϕ ψ, N u,v, ϕ ψ ) E (x1,x ) ( N u1,v 1, ϕ ψ N u,v, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R Tu α1 1 u (ϕt α 1 u 1 u ϕ)(x 1)µ 1 (dx 1 ) Tv α 1 v (ψt α v 1 v ψ)(x )µ (dx ), (3.1) R where T α i t denotes the semigroup of the symmetric α i -stable process. Using an expansion of the characteristic function (see e.g., [5], p. 97) in the integrand with respect to (x 1, x ) in (3.8), it is equal to 1 + iθ ( k ) ( k ) E (x1,x F ) d j L T s j,t j, ϕ ψ θ T F E (x1,x ) d j L T s j,t j, ϕ ψ j1 T j1 where + δ T (x 1,x ), Since k d j E( L T s j,t j, ϕ ψ ) j1 then (3.8) becomes R ( k 3 δ(x T 1,x ) θ3 FT 3 E (x1,x ) d j L T s j,t j, ϕ ψ ). (3.11) j1 R R j1 { [ ( θ k C T (θ) exp R R FT E (x1,x ) k d j E (x1,x )( L T s j,t j, ϕ ψ )µ 1 (dx 1 )µ (dx ), + δ T (x 1,x ) j1 ) d j L T s j,t j, ϕ ψ ] µ 1 (dx 1 )µ (dx ) and by (3.3) and a previous calculation, ( k 1 d j L T s,t, ϕ ψ ) µ 1 (dx 1 )µ (dx ) 1 F T R R FT E (x1,x ) j1 k j1 d j j 1 k j 1 d j F (1) T j1 T sj T tj T sj T tj R R E (x1,x ) ( N u1,v 1, ϕ ψ N u,v, ϕ ψ ) dv du dv 1 du 1 µ 1 (dx 1 )µ (dx ) k k 1 T sj T sj d j d j T α 1 u 1 u (ϕt α 1 u ϕ)(x dx 1 1 u 1)du du 1 x 1 γ 1 1 F () T R T tj T tj R } (3.1) T α v 1 v (ψt α v 1 v ψ)(x )dv dv 1 dx x γ. (3.13)

9 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 9 Now, recalling (3.6) we have 1 ( ) T 1 (1+γ)/α 1 ( ) T 1 (1+γ)/α T s1 T s R T s1 T s R Tu α 1 u (ϕt u α 1 u ϕ)(x)du dx du 1 x γ R R p α u 1 u (x y)ϕ(y) p α u 1 u (y z)ϕ(z)dzdydu du 1 dx x γ, substituting u 1 T u 1, u T u, using the self-similarity of the α-stable process in R, i.e., p α t (x) t 1/α p α 1 (t 1/α x), and then substituting x (T (u 1 u )) 1/α x, s1 s T (1+γ)/α p α T (u 1 R R u )(x y)ϕ(y) p α T u 1 R u (y z)ϕ(z)dzdydu du dx 1 x γ s1 s T (γ 1)/α (u 1 u ) 1/α u 1 u 1/α R ( ) p α 1 (T (u 1 u )) 1/α (x y) ϕ(y) R ( ) p α 1 (T u 1 u ) 1/α (y z) ϕ(z)dzdydu du dx 1 R x γ s1 s ( ) (u 1 u ) γ/α u 1 u 1/α p α 1 (x (T (u 1 u )) 1/α y) ϕ(y) R ( p α 1 R (T u 1 u ) 1/α (y z) R ) ϕ(z)dzdydu du dx 1 x γ. (3.14) Taking T in (3.14) we obtain the limit ( ) p α p α 1 (x) s1 s 1 () ϕ(y)dy R R x γ dx (u 1 u ) γ/α u 1 u 1/α du du 1 ( p α 1 () 1 1 1/α ϕ(y)dy R s1 s ) R p α 1 (x) x γ dx u γ/α [ (s 1 u) 1 1/α + (s u) 1 1/α] du. (3.15) By (3.13), (3.14) and (3.15), ( k 1 d j L T s j,t j ; ϕ ψ ) µ 1 (dx 1 )µ (dx ) lim T F T E (x1,x ) R R pα1 1 () p α 1 () 1 1/α 1 1 1/α ( j1 ϕ(y)dy R ) ( ) ψ(y)dy R R 1 (x) x dx γ1 R p α1 p α 1 (x) x dx γ

10 1 JOHANNA GARZÓN k j,j 1 d j d j sj s j u γ 1/α 1 [(s j u) 1 1/α1 + (s j u) 1 1/α1 ]du tj t j v γ/α [(t j v) 1 1/α + (t j v) 1 1/α ]dv D k d j d jc (1) (s j, s j )C () (t j, t j ), (3.16) j,j 1 where D is defined by (3.7) and C (i) is as in (1.) with a i γ i /α i, b i 1 1/α i. Proceeding similary with the third order term we find ( k 3 1 d j L sj,t j, ϕ ψ ) µ 1 (dx 1 )µ (dx ) R R FT 3 E (x1,x ) 1 F 3 T 1 F 3 T k i1 k k j1 d i d j i1 j1 l1 k i1 d i d i k j1 k j1 d j d j k l1 k d l R R E (x1,x )( L T s i,t i, ϕ ψ L T s j,t j, ϕ ψ L T s l,t l, ϕ ψ )µ 1 (dx 1 )µ (dx ) k T si T ti T sj T tj T sl d l l1 d l 1 F (1)3 T R R 1 T ti T tj T tl F ()3 T R T tl E (x1,x )( N ui,v i, ϕ ψ N u,v, ϕ ψ N u3,v 3, ϕ ψ ) dv 3 du 3 dv du dv 1 du 1 µ 1 (dx 1 )µ (dx ) R T si T sj T sl T α 1 ũ 1 ϕ(t α 1 ũ ũ 1 ϕ(t α 1 ũ 3 ũ ϕ))(x 1 ) dũ 3 dũ dũ 1 dx 1 x 1 γ1 dx T α ṽ 1 ψ(t α ṽ ṽ 1 ψ(t α ṽ 3 ṽ ψ))(x )dṽ 3 dṽ dṽ 1 x γ, (3.17) ũ 1, ũ, ũ 3 denoting u 1, u, u 3 in increasing order, and similarly for ṽ 1, ṽ, ṽ 3. Again, recalling (3.6), substituting ũ i T ũ i i 1,, 3, using self-similarity of the α-stable process, and then substituting x (T ũ 1) 1/α x, we have 1 T s1 T s T s3 ( ) T 1 (1+γ)/α 3 Tũ α 1 ϕ(tũ α ũ 1 ϕ(tũ α dx 3 ũ ϕ))(x)dũ 3 dũ dũ 1 x γ T (γ 1)/α R R R R s1 s s3 p α 1 (x (T ũ 1 ) 1/α w)ϕ(w) R ũ γ/α 1 (ũ ũ 1 ) 1/α (ũ 3 ũ ) 1/α p α 1 ((T (ũ ũ 1 )) 1/α (w y))ϕ(y) p α 1 ((T (ũ 3 ũ )) 1/α (y z))ϕ(z)dzdydwdũ 3 dũ dũ 1 dx R x γ, (3.18)

11 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 11 then from (3.17) and (3.18), ( k 3 1 d j L T s j,t j, ϕ ψ ) µ 1 (dx 1 )µ (dx ) where F 3 T E (x1,x ) R R T (γ i 1)/α i i1 A T (x 1, x ) R s1 s s3 j1 R R p α1 1 (x 1 (T ũ 1 ) 1/α 1 w)ϕ(w) R t1 t t3 R R A T (x 1, x )µ 1 (dx 1 )µ (dx ), (3.19) ũ γ 1/α 1 1 (ũ ũ 1 ) 1/α1 (ũ 3 ũ ) 1/α1 R p α1 1 ((T (ũ ũ 1 )) 1/α 1 (w y))ϕ(y) p α 1 1 ((T (ũ 3 ũ )) 1/α1 (y z))ϕ(z)dzdydwdu 3 du du 1 R ṽ γ/α 1 (ṽ ṽ 1 ) 1/α (ṽ 3 ṽ ) 1/α p α 1 (x (T ṽ 1 ) 1/α w)ψ(w) R p α 1 ((T (ṽ ṽ 1 )) 1/α (w y))ψ(y) p α 1 ((T (ṽ 3 ṽ )) 1/α (y z))ψ(z)dzdydwdv 3 dv dv 1. (3.) From (3.) we obtain lim A T (x 1, x )µ 1 (dx 1 )µ (dx ) T R R s1 s s3 t1 t t3 ( ϕ(x)dx R ũ γ 1/α 1 1 (ũ ũ 1 ) 1/α 1 (ũ 3 ũ ) 1/α 1 du 3 du du 1 ṽ γ /α 1 (ṽ ṽ 1 ) 1/α (ṽ 3 ṽ ) 1/α dv 3 dv dv 1 ) 3 ( ) 3 ψ(x)dx R i1 (p αi 1 ()) R p α i 1 (x) x γ dx. (3.1) i Then, from (3.11), (3.19) and (3.1) we get δ(x T 1,x ) µ 1(dx 1 )µ (dx ). (3.) lim T R R Finally, putting (3.1), (3.16) and (3.) together we obtain k k } lim C T (θ) exp { θ d j d j C (1) (s j, s j )C () (t j, t j ), T D j1 j 1 and convergence of finite-dimensional distributions of X T to finite-dimensional distributions of weighted fractional Brownian sheet DW has been proved. Theorem 3.. Under the hypotheses of Theorem 3.1, if γ 1 γ, then X T converges in law to DW in the space of continuous functions C([, τ] [, τ], R)

12 1 JOHANNA GARZÓN for any τ > as T, where W is fractional Brownian sheet with parameters (1 1 α 1, 1 1 α ), and [ 1/ 1 D ϕ(x)dx ψ(x)dx p α i i ()]. R R 1 1/α i1 i Proof. By Theorem 3.1 we have convergence of finite-dimensional distributions of X T to those of DW. It remains to show that the family {X T } is tight. Since these processes are null on the axes, by the Bickel-Wichura theorem [4] we only need prove that there exist even m and positive constants C m, δ 1, δ such that mδ 1, mδ > 1 and sup E (( s1,t 1 X T (s, t )) m ) C m (s s 1 ) mδ 1 (t t 1 ) mδ, for all s 1 < s, t 1 < t. T (3.3) From (3.4), s1,t 1 X T (s, t ) 1 F T then, by (3.1), E(( s1,t 1 X T (s, t )) ) 1 FT 1 FT T s T t T s 1 T s T t T s T t T s 1 T t 1 T s 1 T s T t T s T t T s 1 1 T t 1 T (1+γ1)/α1 1 T (1+γ)/α T s 1 T t 1 T s T s R T t 1 ( N u,v, ϕ ψ E( N u,v, ϕ ψ )) dudv, Cov ( N u1,v 1, ϕ ψ, N u,v, ϕ ψ ) dv du dv 1 du 1 T t 1 T α 1 u 1 u (ϕt α1 u ϕ)(x 1 u 1)dx 1 R T s 1 T s 1 R T t T t T t 1 T t 1 T α v 1 v (ϕt α v 1 v ϕ)(x )dx dv du dv 1 du 1 T α 1 u 1 u (ϕt α1 u 1 u ϕ)(x 1)dx 1 du du 1 R T α v 1 v (ϕt α v 1 v ϕ)(x )dx dv dv 1 E( X (1) T (s ) X (1) T (s 1), ϕ )E( X () T (t ) X () T (t 1), ψ ), (3.4) where X (i) T, i 1,, are occupation time fluctuation processes of independent systems of particles moving in R according to symmetric α i -stable processes with initial configurations given by Poisson random measures on R with intensities µ i, i.e., and X (1) T (t) 1 T 1 1/α1 X () T (t) 1 T 1 1/α T t T t ( N (1) u, ϕ E( N (1) u, ϕ ))du, N (1) u x Pois(µ 1 ) ( N () u, ϕ E( N () u, ϕ ))du, N () u x Pois(µ ) δ ξ x u, δ ζ x u.

13 CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 13 In [6] such a one-time system is studied and it is shown that E( X (i) T (t) X(i) T (s), ϕ ) C t s h, (3.5) where C is a positive constant (not depending on T ) and h 1 α i > 1. From (3.4) and (3.5) we obtain (3.3). Remark 3.3. We make some remarks below. (1) Theorem 3. gives a functional approximation of fractional Brownian sheet with parameters (h 1, h ) (1/, 3/4], taking h i 1 1 α i, i 1,. () Proving tightness with γ 1 or γ is considerably more difficult because it requires computing moments of arbitrarily high order (see [8] for the one time case), and this involves moments of arbitrarily high order of the Poisson random measure Pois(µ 1 µ ), which are cumbersome. (3) In Theorem 3.1 we may also consider the measures µ i of the form (3.5) with γ i <, assuming that γ i < α i if α i < (which implies that the mean is finite), and the result in the theorem holds. (4) The role of the functions ϕ, ψ is only subsidiary since they are fixed, and in the occupation time fluctuation limit they appear only in the constant D given by (3.7). If ϕ, ψ are taken as variables in the space S(R) of smooth rapidly decreasing functions, then in principle it is possible to prove convergence of the occupation time fluctuations as (S (R)) -valued processes, where S (R) is the space of tempered distributions (topological dual of S(R)), the limit being the space-time random field (Z s,t ) s,t K(λ λ) (W s,t ) s,t, where W is the weighted fractional Brownian sheet in Theorem 3.1, ( ) 1/ 1 K p αi 1 1 1/α () p αi 1 (x) dx, i x γi i1 and λ is the Lebesgue measure on R. one-time particle system.) R References (See [8] for such a setup for a 1. Ayache, A., Leger S., and Pontier, M.: Drap brownien fractionnaire, Potential Anal. 17 () Bardina, X., Jolis, M., and Tudor, C. A.: Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes, Stat. Probab. Lett. 65 (3) Bardina, X. and Florit, C.: Approximation in law to the d parameter fractional Brownian sheet based on the functional invariance principle, Rev. Mat. Iberoamericana 1 (5) Bickel, P. and Wichura, M.: Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist. 4 (1971) Billingsley, P.: Probability and Measure, Wiley, Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Limits theorems for occupation time fluctuations of branching systems: I Long-range dependence, Stoch. Proc. Appl. 116 (6) 1 18.

14 14 JOHANNA GARZÓN 7. Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems, Elect. Comm. in Probab. 1 (7) Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Occupation time limits of inhomogeneous Poisson systems of independent particles, Stoch. Proc. Appl. 118 (8) Coutin, L. and Pointer, M.: Approximation of the fractional Brownian sheet via Ornstein- Uhlenbeck sheet, ESAIM PS 11 (7) Feyel, D. and De La Pradelle, A.: Fractional integrals and Brownian processes, Potential Anal. 1 (1991) Kallenberg, O.: Random Measures, Akademie-Verlag, Berlin (1986). 1. Kamont, A.: On the fractional anisotropic Wiener field, Prob. Math. Statist. 16 (1996) Kühn, T. and Linde, W.: Optimal series representations of fractional Brownian sheets, Bernoulli 8 () Nzi, M. and Mendy, I.: Approximation of fractional Brownian sheet by a random walk in anisotropic Besov space, Random Oper. Stoch. Equ. 15 (7) Tudor, C. A.: Weak convergence to the fractional Brownian sheet in Besov spaces, Bulletin of the Brazilian Mathematical Society 34(3) (3) Johanna Garzón: Department of Mathematics, CINVESTAV, Mexico City, Mexico address: johanna@math.cinvestav.mx

15 Communications on Stochastic Analysis Vol. 3, No. 1 (9) Serials Publications P AC COMMUTATORS AND THE R TRANSFORM AUREL I. STAN Abstract. We develop an algorithmic method for working out moments of a probability measure on the real line from the preservation annihilation creation operator commutator relations for the measure. The method is applied to prove a result of Voiculescu on the R transform. 1. Introduction Expanding on a program introduced in [1] and continued in [], it was proven in [8] that the moments of a probability distribution can be recovered from the commutator between its annihilation and creation operators, and the commutator between its annihilation and preservation operators, provided that the first order moment is given. Moreover, a simple, concrete method for computing the moments was introduced in [8]. In the present paper we apply this method to some classic distributions and to give another proof of a theorem of Voiculescu concerning the R transform. There are already some known techniques for recovering the moments or even the probability distribution of a random variable X, having finite moments of all orders, from its Szegö Jacobi parameters. One method uses a continued fraction expansion of the Cauchy Stieltjes transform of X and is very useful when the random variable has a compact support. Another powerful way is the method of renormalization introduced in [3, 5, 4] and pushed almost to the limits in [7]. However, our method is based on the Lie algebra structure of the algebra generated by the annihilation, preservation, and creation operators. In section we introduce very quickly the annihilation, preservation, and creation operators for a one dimensional distribution having finite moments of any order. For brevity we will call these operators the P AC operators. We also present the commutator method and its dual developed in [8]. In section 3 we apply this method to two families of distributions. Finally, in section 4, we use the commutator method and its dual to give a proof of an important theorem, of Voiculescu, about the analytic function theory tools for computing the R transform.. Background Let X be a random variable having finite moments of any order, i.e., E[ X p ] <, for all p >, where E denotes the expectation. It is well known that by applying the Gram Schmidt orthogonalization procedure to the sequence of monomial Mathematics Subject Classification. Primary 81S5; Secondary 5E35. Key words and phrases. Moments, Szegö Jacobi parameters, creation, annihilation, preservation, commutator, Cauchy Stieltjes transform, R transform. 15

16 16 AUREL I. STAN random variables: 1, X, X,..., we can obtain a sequence of orthogonal polynomial random variables: f (X), f 1 (X), f (X),..., chosen such that for each n, f n has degree n and a leading coefficient equal to 1. We assume that the probability distribution of X has an infinite support so that the sequence f, f 1, f n,... never terminates. There exist two sequences of real numbers: {α k } k and {ω k } k 1, called the Szegö Jacobi parameters of X, such that for all n, we have: Xf n (X) f n+1 (X) + α n f n (X) + ω n f n 1 (X). (.1) See [6] and [9]. When n, in this recursive relation f 1 (the null polynomial) and ω : by agreement. The terms of the sequence {ω k } k 1 are called the principal Szegö Jacobi parameters of X and they must all be positive since, for all n 1, E [ f n (X) ] ω 1 ω ω n. (.) Moreover, by Favard s theorem, given any sequence of real numbers {α k } k and any sequence of positive numbers {ω k } k 1, there exists a random variable X having these sequences as its Szegö Jacobi parameters. Let F be the space of all random variables of the form f(x), where f is a polynomial function, and for each n, let F n be the subspace of F consisting of all random variables f(x), such that f is a polynomial of degree at most n. Let G : F, and for all n 1, let G n : F n F n 1, i.e., the orthogonal complement of F n 1 into F n. For each n, G n Cf n (X) (scalar multiples of f n (X)) and G n is called the homogenous chaos space of degree n generated by X. The space H : n G n is called the chaos space generated by X. It is clear that F is dense in H. For each n, we denote by P n the orthogonal projection of H onto G n. If we look back to the recursive formula (.1), then we can easily see that, for all n : and P n+1 [Xf n (X)] f n+1 (X), P n [Xf n (X)] α n f n (X), P n 1 [Xf n (X)] ω n f n 1 (X). Let us regard now X not as a random variable, but as a multiplication operator from F to F, which maps a polynomial random variable f(x) into Xf(X). We can see that applying the multiplication operator X to a polynomial from G n we get three polynomials: one in G n+1, one in G n, and one in G n 1. That means: X G n P n+1 X G n + P n X G n + P n 1 X G n. We define D n + : P n+1 X G n, Dn : P n X G n, and Dn : P n 1 X G n. Since D n + maps G n into G n+1, it increases the degree of f n, and thus it is called a creation operator. Similarly, since Dn : G n G n, it is called a preservation operator, and since Dn : G n G n 1, it is called an annihilation operator. So far the annihilation, preservation, and creation operators have been defined only on each individual homogenous chaos space G n. We extend their definition as linear operators from F to F, and define the operators: a, a, and a +, such that for

17 P AC COMMUTATORS AND THE R TRANSFORM 17 any n, a G n : D n, a G n : D n, and a + G n : D + n. As a multiplication operator X is the sum of these three operators. Thus: X a + a + a +. (.3) It is known that X is polynomially symmetric, i.e., E[X k 1 ], for all positive integers k, if and only if a, see [1], or equivalently α n, for all n. In this case X a + a +. We will briefly explain now the commutator method introduced in [8], used to recover the moments and if possible the probability distribution of a random variable X, from the commutator between its annihilation and creation operators, commutator between its annihilation and preservation operators, and its first moment, E[X]. We would like to make the reader aware of the fact that some times we regard X as a random variable, and other times we view it as a multiplication operator. We hope that this will not create any confusion, since most of the time, when we refer to it as being a multiplication operator, we will write X n 1, where 1 is the constant (vacuum) polynomial equal to 1. When we write E[X n ], for some n 1, we regard X as a random variable. The commutator of two operators A and B is defined as: [A, B] : AB BA. (.4) Commutator Method Let X be a random variable, having finite moments of all orders. We assume that [a, a + ], [a, a ], and E[X] are given (known). Then, in order to compute the higher moments of X, we will follow the following three steps. Step 1. Let, denote the inner product defined as: f(x), g(x) : E[f(X)g(X)], for all polynomials f and g. For any fixed positive integer n, we have: E[X n ] XX n 1 1, 1 (a + a + a + )X n 1 1, 1 a X n 1 1, 1 + a X n 1 1, 1 + a + X n 1 1, 1. Since (a ) a, (a + ) a, a 1 E[X]1, and a 1, we have: and Thus a + X n 1 1, 1 X n 1 1, a 1 X n 1 1, a X n 1 1, 1 X n 1 1, a 1 X n 1 1, E[X]1 E[X] X n 1 1, 1 E[X]E[X n 1 ]. E[X n ] E[X]E[X n 1 ] + a X n 1 1, 1.

18 18 AUREL I. STAN Step. Swap (permute) a and X n 1, using the simple formula: and the product rule for commutators: [A, B k ] AB BA + [A, B] k 1 B k 1 j [A, B]B j, for all operators A and B, and any k. Use also the fact that Thus we get: j [a, X] [a, a + a + a + ] [a, a ] + [a, a ] + [a, a + ] [a, a ] + [a, a + ]. E[X n ] E[X]E[X n 1 ] + a X n 1 1, 1 E[X]E[X n 1 ] + ( X n 1 a + [a, X n 1 ] ) 1, 1 E[X]E[X n 1 ] + X n 1 a 1, 1 + [a, X n 1 ]1, 1 E[X]E[X n 1 ] + [a, X n 1 ]1, 1 n E[X]E[X n 1 ] + X n j [a, X]X j 1, 1 j n E[X]E[X n 1 ] + X n j ( [a, a ] + [a, a + ] ) X j 1, 1. j Step 3. If necessary, go back to Step and repeat the procedure, until a recursive formula expressing the n th moment in terms of lower order moments is obtained. The idea in this method (algorithm) is very simple: move each annihilator a stepwise to the right, using the commutator relations with a + and a, until it acts on the vacuum polynomial 1 and kills it. There is also a dual of this method, using the creation operator a +, instead of the annihilation operator a. We will briefly explain it now. Dual Commutator Method Step 1. For any fixed positive integer n, we have: E[X n ] X n 1 X1, 1 X n 1 (a + a + a + )1, 1 X n 1 a 1, 1 + X n 1 a 1, 1 + X n 1 a + 1, 1 + X n 1 E[X]1, 1 + X n 1 a + 1, 1 E[X]E[X n 1 ] + X n 1 a + 1, 1. Step. Swap X n 1 and a +. Step 3. Repeat Step if necessary. In the dual commutator method the creation operator a + is moved stepwise to the left, until it arrives to the left most possible position. In that moment, for any

19 P AC COMMUTATORS AND THE R TRANSFORM 19 polynomial f, we have: a + f(x)1, 1 f(x)1, a Some Calculations In this section we apply the commutator method to two concrete examples. Example 3.1. Let us consider now a random variable X, having finite moments of all orders, whose Szegö Jacobi parameters are α n, for all n, and the principal Szegö Jacobi parameters are: c, c + d, c + d, c + d, 3c + d, 3c + 3d,.... That means, for all n 1, ω n 1 nc + (n 1)d and ω n nc + nd, where c and d are fixed real numbers, such that c > and c + d >. Since α n, for all n, we know that X is symmetric and thus the space spanned by the monomial random variables of even degree: 1, X, X 4,..., is orthogonal to the space spanned by the monomial random variables of odd degrees: X, X 3, X 5,.... In fact the closures of these two spaces are H e : G G G 4 and H o : G 1 G 3 G 5. Let E : H H e and O : H H o denote the orthogonal projections of H onto H e and H o, respectively. Since, for all n, [a, a + ]f n (X) (ω n+1 ω n )f n (X), where {f n } n are the orthogonal polynomials generated by X, we can see that the commutator of the annihilation and creation operators is: [a, a + ] ce + do. (3.1) Because a, all the odd moments vanish. Applying now our commutator method, for all n 1, we have: E[X n ] X n j [a, a + ]X j 1, 1 n j n c X n j EX j 1, 1 + d j X n j OX j 1, 1. n Since EX k X k, EX k+1, OX k+1 X k+1, and OX k, for all k, we get: n 1 n E[X n ] c X n k X k 1, 1 + d X n k 1 X k+1 1, 1 k k j cne[x n ] + d(n 1)E[X n ] [(c + d)n d]e[x n ]. Iterating this recursive relation, we obtain: E[X n ] [(c + d)n d]e[x n ] [(c + d)n d][(c + d)(n 1) d]e[x n 4 ]... [(c + d)n d][(c + d)(n 1) d] [(c + d)1 d]e[x ].

20 AUREL I. STAN Thus we obtain that: 1 (c + d) n E[Xn ] ( n d ) ( n 1 c + d d ) (... 1 d ). (3.) c + d c + d We recognize that the right hand side of (3.) is exactly the n th moment of a gamma distribution. That means the distribution of the random variable Y : [1/(c + d)]x is given by the density function: f(x) 1 ( Γ c c+d )x c c+d 1 e x 1 (, ) (x), (3.3) where Γ denotes the Euler s gamma function. Thus X is a re scaled gamma random variable. Since X is a symmetric random variable, we can compute first its distribution function F X in the following way: F X (a) : P (X a) 1 P (X > a) 1 1 P (X > a ) P ([1/(c + d)]x a /(c + d)) F Y (a /(c + d)), for all a >. Differentiating both sides of the last equality with respect to a, we obtain that the density of X is: g(a) F X(a) a c + d F Y (a /(c + d)) a c + d f(a /(c + d)), for all a >. Since g( a) g(a), we conclude that X is the random variable given by the density function: g(x) 1 ( (c + d) c c+d Γ c c+d ) x c d c+d e x c+d. (3.4) Example 3.. Let us find now the random variable X whose Szegö Jacobi parameters are: { α if n α n if n 1 and ω n { b if n 1 c if n, where α, b, and c are fixed real numbers, such that b and c are strictly positive.

21 P AC COMMUTATORS AND THE R TRANSFORM 1 Before computing the moments of X, we will find a simple upper bound for E[ (X α) n ], for each n. Claim 1. For each n, we have: E[ (X α) n ] (3T ) n, (3.5) where T : max{1, α, b, c}. Indeed, let n be fixed. Let {f n } n denote the sequence of orthogonal polynomials, with a leading coefficient equal to 1, generated by X. We have: E[(X α) n ] (X αi) (X αi)1, 1 {a + a + + (a αi)} {a + a + + (a αi)}1, 1 a 1 a n 1, 1. (a 1,,a n ) {a,a +,a αi} n Observe that in the last sum only the terms that contain the same number of annihilation and creation operators could be non zero, since we start from the vacuum space R1 and we have to return to this space (otherwise a 1 a n 1 1). For these terms, we move from one orthogonal polynomial to another in the following way. If a j a +, and we are currently at f k, then a j f k f k+1, and we retain a coefficient c j 1. If a j a, then a j f k ω k f k 1 and we retain a coefficient c j ω k. Finally, if a j a αi, then a j f k (α k α)f k and we retain a coefficient c j α k α. Observe, that for all j, we have c j T. Since at the end we return to the vacuum polynomial 1, we get: E[(X α) n ] c 1 c n 1, 1 c 1 c n c 1 c n 3 n T n, since the cardinality of the set {a, a, a αi} n is 3 n. Using now Jensen s inequality we get: E[ X n ] E[X n ] (3T ) n. Let us compute now the moments of X. We have: [a, a + ]f n (ω n+1 ω n )f n. (3.6) Thus [a, a + ]f bf, [a, a + ]f 1 (c b)f 1, and [a, a + ]f n, for all n. This means that: [a, a + ] bp + (c b)p 1, (3.7) where P k denotes the projection onto the space G k Cf k. Moreover, since [a, a ]f n (α n α n 1 )ω n f n 1, (3.8) for all n, where α 1 :, we conclude that: [a, a ] αa P 1. (3.9)

22 AUREL I. STAN From the recursive relation: Xf (X) f 1 (X) + α f (X) + ω f 1 (X), since f 1, we conclude that f 1 (X) X α. Moreover E[f 1 (X) ] ω 1 b. Thus {1} and {(1/ b)(x α)} are orthonormal bases of G and G 1, respectively. Hence for all polynomial functions g, we have: and P g(x) g(x), 1 1 E[g(X)]1 P 1 g(x) g(x), (1/ b)(x α) (1/ b)(x α) 1 g(x), X α (X α) b 1 E[(X α)g(x)](x α). b We apply now our commutator method to compute the moments of X. Actually, it is easier to compute the moments of X α than those of X. For any fixed natural number n, we have: E[(X α) n ] (a + + a + a αi)(x αi) n 1 1, 1 a + (X αi) n 1 1, 1 + (a αi)(x αi) n 1 1, 1 + a (X αi) n 1 1, 1. Here I denotes the identity operator of H. We have: and Thus we have: a + (X αi) n 1 1, 1 (X αi) n 1 1, a 1 (a αi)(x αi) n 1 1, 1 (X αi) n 1 1, (a αi)1. E[(X α) n ] a (X αi) n 1 1, 1. We swap now a and (X αi) n. Since after the swap the annihilation operator a kills the vacuum polynomial 1, we get: E[(X α) n ] [a, (X αi) n 1 ]1, 1.

23 P AC COMMUTATORS AND THE R TRANSFORM 3 Thus we obtain: E[(X α) n ] [a, (X αi) n 1 ]1, 1 + n (X αi) n j [a, X αi](x αi) j 1, 1 j n (X αi) n j [a, a + + a + a αi](x αi) j 1, 1 j n (X αi) n j [a, a + ](X αi) j 1, 1 j n (X αi) n j [a, a ](X αi) j 1, 1 j n (X αi) n j [bp + (c b)p 1 ](X αi) j 1, 1 j (X αi) n j a P 1 (X αi) j 1, 1. n α j Since P (X α) j E[(X α) j ]1 and P 1 (X α) j (1/b)E[(X α) j+1 ](X α), we obtain the following recursive formula: n E[(X α) n ] b E[(X α) j ]E[(X α) n j ] j + c b b n α n E[(X α) j+1 ]E[(X α) n 1 j ] j E[(X α) j+1 ]E[(X α) n j ], j for all n 1. Multiplying both sides of this recursive relation by t n and then summing up from n 1 to infinity, we obtain that the function ϕ(t) E[1/(1 t(x α))] satisfies the following equation: ϕ(t) 1 bt ϕ (t) + c b [ϕ(t) 1] αtϕ(t)[ϕ(t) 1], b for all t in a neighborhood of. It must be observed, that in deriving this formula we interchanged the summation with the expectation, which is possible for the small values of t, due to the inequality (3.5). This relation is equivalent to the quadratic equation in ϕ(t): (bt αt + p 1)ϕ (t) + (αt p + 1)ϕ(t) + p, (3.1)

24 4 AUREL I. STAN where p : c/b. Using the quadratic formula, we get: ϕ(t) αt + p 1 ± (αt + 1) 4pbt (bt αt + p 1) αt + p 1 ± (αt + 1) 4pbt (bt αt + p 1) αt + p 1 (αt + 1) 4pbt αt + p 1 (αt + 1) 4pbt 4p ( bt αt + p 1 ) (bt αt + p 1) ( αt + p 1 ) (αt + 1) 4pbt Since, ϕ() E[1] 1, we get: p αt + p 1 (αt + 1) 4pbt. ϕ(t) p αt + p 1 + (αt + 1) 4pbt, for all t in a neighborhood of. Thus we get [ ] 1 p E 1 t(x α) αt + p 1 + (αt + 1) 4pbt. (3.11) Replacing t by 1/t, we obtain that the Cauchy Stieltjes transform of X is: [ ] 1 p E t (X α) α + (p 1)t + s(t) (t + α) 4pb, (3.1) for all t away from, where s(t) denotes the sign function of t, i.e., s(t) t/ t. We can invert the Cauchy Stieltjes transform to find the probability distribution of X α first, and then of X. We are not going over this computation, but the interested reader can read Theorem 5.3 from [7]. 4. The R transform We will close the paper, by giving a new proof of a theorem by Voiculescu concerning the analytic function theory tools for computing the R transform. We will briefly explain this transform, following the concepts from [1]. Let H Ce be a one dimensional Hilbert space, where {e} is an orthonormal basis of H. Let Γ(H) be the full Fock space generated by H, that means: Γ(H) : C1 H H H 3, where means orthogonal direct sum. We define a left creation operator a + (denoted by l in [1]) on Γ(H) in the following way: { a + e if τ 1 τ (4.1) e τ if τ Γ(H) C1,

25 P AC COMMUTATORS AND THE R TRANSFORM 5 where denotes the orthogonal complement. The adjoint of this operator is the left annihilation operator a (denoted by l in [1]) and is defined as: { a k1, e k (k 1 k k n ) k n if n 1 (4.) if n, where for n, k 1 k k n is understood to be a complex multiple of 1 (that means an element of C1), and, denotes the inner product of H. It is not hard to see that the commutator of the left creation and annihilation operators is: [a, a + ] P, (4.3) where P denotes the orthogonal projection of Γ(H) onto the vacuum space C1. Moreover, a a + I, where I denotes the identity operator of Γ(H). Definition 4.1. A noncommutative probability space is a unital algebra, A over C together with a linear functional, φ : A C, such that φ(1) 1. Every element f in A is called a random variable, and φ(f) is called the expectation of f. For this reason we will replace the letter φ from [1] by E. Every random variable f from A generates a distribution µ f on the algebra of complex polynomials in one variable C[X], i.e., a linear functional from C[X] to C that maps the constant polynomial 1 into the complex number 1. It is defined by the formula: µ f (P [X]) : E[P (f)], (4.4) for all P [X] C[X]. Let Σ denote the set of all linear functionals µ, on C[X], such that µ(1) 1. Proposition 4.. For all p 1, p, q 1, and q non-negative integers, we have: if and only if p 1 p and q 1 q. (a + ) p1 (a ) q1 (a + ) p (a ) q (4.5) Proof. Let us assume that (a + ) p 1 (a ) q 1 (a + ) p (a ) q. Since for all k q 1, (a + ) p 1 (a ) q 1 maps H k into H (k+p 1 q 1 ), and for all k q, (a + ) p (a ) q maps H k into H (k+p q ), we conclude that: p 1 q 1 p q. (4.6) Let us assume that p p 1 and thus, m : p p 1 q q 1. By composing (a ) p 1 with each side of the equality (4.5), we get: (a ) p1 (a + ) p1 (a ) q1 (a ) p1 (a + ) p (a ) q. Since a a + I and p p 1, we obtain: which means: (a ) q 1 (a + ) p p 1 (a ) q, (a ) q1 (a + ) m (a ) q. (4.7)

26 6 AUREL I. STAN Let us compose now each side of the equality (4.7) with (a + ) q 1 (to the right). We obtain: Since a a + I and q q 1, we have: which means: (a ) q 1 (a + ) q 1 (a + ) m (a ) q (a + ) q 1. I (a + ) m (a ) q q 1, I (a + ) m (a ) m. (4.8) If m >, then the equality (4.8) is impossible since if we apply each side of it to the vacuum vector 1, we get: I1 1 while (a + ) m (a ) m 1, because a kills the vacuum vector. Thus m and so, p 1 p and q 1 q. Proposition 4. allows us to define the unital algebra Ẽ1 of formal series of the form: Q c p,q (a + ) p (a ) q, (4.9) q p where Q and c p,q C, for all q Q and p. Ẽ 1 is an algebra (i.e., closed under multiplication) due to the fact that the creation operators are always to the left of the annihilation operators, and a a + I. We define the expectation E (i.e., the linear functional φ mapping 1 to 1), on Ẽ, in the following way: [ Q ] E c p,q (a + ) p (a ) q : c,. (4.1) q p Let us observe that formally, for any f Ẽ1, we have: E[f] f1, 1, (4.11) where, denotes the inner product of the Fock space Γ(H) and 1 the vacuum vector of Γ(H). Thus (Ẽ, E) is a noncommutative probability space. Voiculescu proved (see [1]) that, for every µ Σ (let us remember that Σ denotes the set of all linear functionals µ, on C[X], such that µ(1) 1), there exists a unique random variable, T µ, of the form a + k α k+1(a + ) k in Ẽ1, whose distribution in (Ẽ, E) is µ. Here the numbers α 1, α,..., represent arbitrary coefficients and have nothing to do with the Szegö Jacobi parameters. T µ is called the canonical random variable of µ. We define the R transform of µ to be the formal power series: R µ α k+1 x k. (4.1) k We will give a proof of the following theorem (Theorem from [1]), using both our commutator and dual commutator method.

27 P AC COMMUTATORS AND THE R TRANSFORM 7 Theorem 4.3. Let µ be a distribution on C[X], with R transform R µ (z) α k+1 z k. (4.13) Then denoting by µ k the kth moment of µ, µ(x k ), we have that the formal power series G(w) w 1 + µ k w k 1 (4.14) and are inverses with respect to composition. k k1 K(z) 1 z + R µ(z) (4.15) Proof. Let T µ : a + k α k+1(a + ) k Ẽ1 be the canonical random variable of µ. Let us compute the moments of µ, or equivalently of T µ, using our commutator method. For all n 1, we have: µ n E [ Tµ n ] Tµ n 1, 1 [ ] a + α k+1 (a + ) k Tµ n 1 1, 1 k a T n 1 µ 1, 1 + α 1 T n 1 µ 1, 1 + k1 α k+1 (a + ) k T n 1 µ 1, 1. For all k 1, (a + ) k T n 1 µ 1 Γ(H) C1, and thus, (a + ) k T n 1 µ 1, 1. Hence, we obtain: We have: µ n a T n 1 µ 1, 1 + α 1 µ n 1 [a, T n 1 µ ]1, 1 + T n 1 µ a 1, 1 + α 1 µ n 1 n T n j j µ [a, T µ ]T j µ1, 1 + α 1 µ n 1. [a, T µ ] [a, a + α 1 I + α k+1 (a + ) k ] k1 α k+1 [a, (a + ) k ] k1 k1 k1 α k+1 k 1 (a + ) k 1 r [a, a + ](a + ) r r α k+1 k 1 (a + ) k 1 r P (a + ) r. r

28 8 AUREL I. STAN Now, we make the crucial observation that, for all r 1, P (a + ) r, due to the fact that the range of (a + ) r is H r H (r+1) which is orthogonal to the vacuum space C1 (the range of P ). Thus in the sum k 1 r (a+ ) k 1 r P (a + ) r, from the commutator [a, T µ ], only the term corresponding to r survives. Therefore, we get: [a, T µ ] α k+1 (a + ) k 1 P. It follows now, that: Since we obtain: µ n In the last sum: j k1 n Tµ n j [a, T µ ]Tµ1, j 1 + α 1 µ n 1 n α 1 µ n 1 + j k1 n µ n α 1 µ n 1 + α k+1 T n j µ (a + ) k 1 P T j µ1, 1. P T j µ1 T j µ1, 1 1 j E[T j µ]1 µ j 1, µ j j k1 α k+1 T n j µ (a + ) k 1 1, 1 n α 1 µ n 1 + α µ j Tµ n j 1, 1 + n µ j α k+1 Tµ n j (a + ) k 1 1, 1 j k n α 1 µ n 1 + α µ j µ n j + j k1 j n µ j α k+ Tµ n j (a + ) k 1, 1. n µ j α k+ Tµ n j (a + ) k 1, 1, j k1 j is actually running from to n 3, since for j n, we have: T n j µ (a + ) k 1, 1 (a + ) k 1, 1 (a + ) k 1 1, a 1,

29 P AC COMMUTATORS AND THE R TRANSFORM 9 for all k 1. We will now use the dual commutator method, to bring the creation operators from right to left. In the last sum: n 3 µ j α k+ Tµ n j (a + ) k 1, 1, j k1 we swap T n j µ and (a + ) k using the commutator formula: [T n j µ, (a + ) k ]1 n 3 j i n 3 j i n 3 j i n 3 j i T n 3 j i µ [T µ, (a + ) k ]T i µ1 T n 3 j i µ [a, (a + ) k ]T i µ1 T n 3 j i µ (a + ) k 1 P T i µ1 µ i T n 3 j i µ (a + ) k 1 1. Thus, since after the swap (a + ) k T n j µ 1, 1, we obtain: n µ n α 1 µ n 1 + α µ j µ n j + n 3 µ j j k1 j n 3 j α k+ i n α 1 µ n 1 + α µ j µ n j + + n 3 j n 3 j µ j α 3 n 3 µ j j k i j µ i T n 3 j i µ (a + ) k 1 1, 1 µ i T n 3 j i µ 1, 1 n 3 j α k+ i n α 1 µ n 1 + α µ j µ n j n 3 + α 3 + j n 3 j i n 3 µ j j k1 j µ j µ i µ n 3 j i n 3 j α k+3 i µ i T n 3 j i µ (a + ) k 1 1, 1 µ i T n 3 j i µ (a + ) k 1, 1.

30 3 AUREL I. STAN We observe, as before, that in the last sum: n 3 µ j j k1 n 3 j α k+3 i µ i T n 3 j i µ (a + ) k 1, 1, j is actually running from to n 4, and i from to n 4 j. We repeat this procedure swapping now Tµ n 3 j i and (a + ) k, and so on, each time reducing the running interval for j, until this interval disappears. It is now clear that in the end, we get: µ n α 1 µ n 1 + α µ j1 µ j + α 3 µ j1 µ j µ j3 + + α n j 1 +j + +j n j 1 +j n j 1 +j +j 3 n 3 µ j1 µ j µ jn, (4.16) for all n 1. Formula (4.16) is very interesting and easy to memorize. Dividing first both sides of formula (4.16) by w n+1, and then summing up from n 1 to, we get: n1 Since µ 1, this means: This is equivalent to: which means: µ n w n+1 1 w α w α + 1 w α 3. G(w) 1 w G(w) w n1 [ n1 [ n1 µ n 1 w n µ n 1 w n µ n 1 w n ] ] 3 α k+1 [G(w)] k k G(w) w R(G(w)). wg(w) G(w)R(G(w)) + 1, w R(G(w)) + 1 G(w) K(G(w)). Thus G(w) and K(z) are inverses with respect to composition. Acknowledgement. The author would like to thank the referee for giving him many important suggestions about how to improve this paper. Thus Claim 1, from Example 3., and Proposition 4. were added to the paper following his/her recommendation.

31 P AC COMMUTATORS AND THE R TRANSFORM 31 References 1. Accardi, L., Kuo, H.-H., and Stan, A. I.: Characterization of probability measures through the canonically associated interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, No. 4 (4) Accardi, L., Kuo, H.-H., and Stan, A. I.: Moments and commutators of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 4 (7) Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functions I, Taiwanese J. Math. 7 (3) Asai, N., Kubo, I., and Kuo, H.-H.: Generating functions of orthogonal polynomials and Szegö Jacobi parameters, Prob. Math. Stat. 3 (3) Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functions II, Taiwanese J. Math. 8 (4) Chihara, T. S.: An Introduction to Orthogonal Polynomials, Gordon & Breach, New York, Namli, S.: Multiplicative Renormalization Method for Orthogonal Polynomials, Ph.D. thesis electonically available at 8. Stan, A. I. and Whitaker, J. J.: A study of probability measures through commutators,. J. Theor. Prob., to appear. 9. Szegö, M.: Orthogonal Polynomials. Coll. Publ. 3, Amer. Math. Soc., Voiculescu, D. V., Dykema, K. J., and Nica, A.: Free Random Variables, Vol. 1, CRM Monograph Series, American Mathematical Society, Providence, Rhode Island USA, 199. Aurel I. Stan: Department of Mathematics, The Ohio State University at Marion, Marion, OH 433, U.S.A. address: stan.7@osu.edu