motion For proofs see Sixth Seminar on Stochastic Analysis, Random Fields and Applications

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1 For proofs see Sixth Seminar on Stochastic Analysis, Random Fields and Applications Division of Applied Mathematics School of Education, Culture, and Communication Mälardalen University SE Västerås, Sweden 1

2 The (FBM) Definition 1 The with the Hurst parameter H (0, 1) is the centred Gaussian random field ξ (x) on the space R N with the covariance function R(x,y) = Eξ (x)ξ (y) = 1 2 ( x 2H + y 2H x y 2H ). (1) 2

3 Let B R = {x R N : x R } be the centred ball of radius R. Consider a mean square converging series representation of the FBM of the form ξ (x) = f α (x)ξ α, x B R, α A where ξ α, α A is the countable set of independent standard normal random variables, and where f α (x) are deterministic. We would like to find a systematic way to build such representations. 3

4 Let m, n, p, and q be four nonnegative integers with 0 m q and 0 n p. Let a 1,..., a p, b 1,..., b q be points in the complex plane. Assume that for each k = 1, 2,..., p and for each i = 1, 2,..., q we have a k b i + 1 / Z + = {0,1,2,...}. Then, there exists an infinite contour L that separates the poles of Γ(1 a k s) at s = 1 a k + j, j Z + from the poles of Γ(b i + s) at s = b i l, l Z +. Definition 2 is defined by ( G m,n p,q z a 1,...,a n,a n+1,...,a p b 1,...,b m,b m+1,...,b q 1 2πi L ) = m j=1 Γ(s + b j ) n j=1 Γ(1 a j s) p j=n+1 Γ(s + a j) q j=m+1 Γ(1 b j s) z s ds. The number p + q is the order of the Meijer G-function. 4

5 The classical Meijer s integral The classical Meijer s integral from two G- is again a G-function! ( τ α 1 G s,t u,w wτ c ) 1,...,c t,c t+1,...,c u 0 d 1,...,d s,d s+1,...,d v ( G m,n p,q zτ a ) 1,...,a n,a n+1,...,a p dτ = w α G m+t,n+s v+p,u+q b 1,...,b m,b m+1,...,b q ( z w a ) 1,...,a n,1 α d 1,...,1 α d v,a n+1,...,a p. b 1,...,b m,1 α c 1,...,1 α c u,b m+1,...,b q 5

6 The Gegenbauer polynomials A plenty of special are specialised values of the Meijer G-function. In particular, the Gegenbauer polynomials C λ n (x) appear as ( G 0,2 2,2 z a,c ) b,b + 1/2 Γ(2b 2c + 2)ϑ( z 1) = Γ(a 2b + c 1/2)(2(a 2b + c 1)) 2b 2c+1 z b (z 1) a 2b+c 3/2 C a 2b+c 1 2b 2c+1 ( z), where (a) n = Γ(a + n)/γ(a) is the Pochhammer symbol, and { 1, x 0, θ(x) = 0, x < 0 is the unit step function. 6

7 The Bessel The Bessel appear as ( J ν (z) = G 1,0 z 2 0,2 4 ν/2, ν/2 ). 7

8 Notation Let m be a nonnegative integer, and let m 0, m 1,..., m N 2 be integers satisfying the following condition m = m 0 m 1 m N 2 0. Let x = (x 1,x 2,...,x N ) be a point in the space R N. Let r k = xk x k x N 2, where k = 0, 1,..., N 2. Consider the following ( xn 1 + ix N H(m k,±,x) = r N 2 C m k+1+(n k 2)/2 m k m k+1 ) ±mn 2 r m N 3 N 2 N 2 ( xk+1 r k k=0 ), r m k m k+1 k and denote Y (m k,±,x) = r m 0 H(m k,±,x). 8

9 Definition of spherical harmonics Definition 3 The Y (m k,±,x) are called the (complex-valued) spherical harmonics. The spherical harmonics are orthogonal in the Hilbert space L 2 (S N 1 ) of the square integrable on the unit sphere S N 1. Let L(m k ) be the square of the length of the vector Y (m k,±,x). Let l = l(m k,± ) be the number of the symbol (m 0,m 1,...,m N 2,±) in the lexicographic ordering. Definition 4 The real-valued spherical harmonics, S l m(x), are defined as Y (m k,+,x)/ L(m k ), m N 2 = 0, Sm(x) l = 2ReY (mk,+,x)/ L(m k ), m N 2 > 0,l = l(m k,+ ), 2ImY (m k,,x)/ L(m k ), m N 2 > 0,l = l(m k, ). 9

10 Notation Note that the ξ (x) is weakly isotropic, i.e., the autocorrelation function (1) is invariant with respect to the group O(N) of the orthogonal matrices of order N. Let t > 0, let S t = {x R N : x = t } be the centred sphere in the space R N, and let dω be the Lebesgue surface measure on S t. Let η(x) be a centred weakly isotropic random field. In fact, the autocorrelation function R(x, y) of the random field η(x) is a function R(s,t,u) of the three real variables s = x, t = y, and u being the cosine of the angle between the vectors x and y. 10

11 Reducing to the case of dimension 1 Yadrenko (1983) proved that the stochastic processes Xm(t) l = η(x)sm(x/ x )dω l S t are centred and uncorrelated. The autocorrelation function of the process X l m(t) is with 1 R m (s,t) = c R(s,t,u)C (N 2)/2 m (u)(1 u 2 ) (N 3)/2 du, 1 c = 2N 2 π (N 2)/2 m!γ((n 2)/2). Γ(m + N 2) The random field η(x) can be represented as η(x) = m=0 h(m,n) Xm( x )S l m(x/ x ), l l=1 where the series converges in mean square for any x R N. 11

12 The autocorrelation function of the process X l m(t) Lemma 1 The autocorrelation function of the stochastic process X0 1(t) has the form ( R 0 (s,t) = c2 NH 2 s2h G 2,2 s 2 4,4 t 2 1 H,0,N/2,1 ), 0,1 H,1 H N/2, H while the autocorrelation function of the stochastic process X l m(t), m 1, has the form ( R m (s,t) = c2 NH 2 sm t 2H m G 1,1 s 2 2,2 t 2 H + 1 m,n/2 + H 0,1 N/2 m ), where c 2 NH = 2π (N 2)/2 Γ(N/2 + H)Γ(H + 1)sin(πH). 12

13 Volterra kernels and Definition 5 Recall that a function a(s,u): (0, ) (0, ) R is called the Volterra kernel, if it is locally square integrable, and a(s,u) = 0 for s < u. Definition 6 A Volterra process with Volterra kernel a(s, u) is a centred Gaussian stochastic process η(t) with autocorrelation function R(s,t) = min{s,t} 0 a(s,u)a(t,u)du. 13

14 X l m(t) are Lemma 2 The stochastic processes Xm(t) l are with Volterra kernels ( ) c NH u H 1/2 G 2,0 u 2 N/2,1 2,2, s 2 m = 0, 0,1 H a m (s,u) = ( ) c NH s 2H m u m H 1/2 G 1,0 u 2 N/2 + H 1,1, s 2 m

15 , slide 1 By Lemma 2, the autocorrelation function of the stochastic process X l m(t) has the form R m (s,t) = min{s,t} 0 a m (s,u)a m (t,u)du. By definition 5, the last display can be rewritten as R m (s,t) = R 0 a m (s,u)a m (t,u)du, s,t [0,R]. For each pair (m,l) with m Z + and 1 l h(m,n), let {e l mn(u): n 1} be a basis in the Hilbert space L 2 [0,R]. We obtain where R m (s,t) = n=1 b l mn(s) = b l mn(s)b l mn(t), R 0 a m (s,u)e l mn(u)du. 1 l h(m,n), 15

16 , slide 2 It follows that the stochastic process X l m(t) itself has the form X l m(t) = n=1 Finally, we have the following result. Theorem b l mn(t)ξ l mn. For any choice of the bases {e l mn(u): n 1}, the ξ (x) has the following series expansion ξ (x) = m=0 h(m,n) l=1 n=1 b l mn( x )S l m(x/ x )ξ l mn. This series converges in mean square for any x in the centred closed ball B R = {x R N : x R }. 16

17 The Fourier Bessel Let ν be a real number, and let j ν,1 < j ν,2 < < j ν,n <... be the positive zeros of the Bessel function J ν (u). For any ν > 1, the Fourier Bessel 2u ϕ ν,n (u) = J ν+1 (j ν,n ) J ν(j ν,n u), n 1 form a basis in the space L 2 [0,1]. By change of variable we conclude that the 2u emn(u) l = RJ ν+1 (j ν,n ) J ν(r 1 j ν,n u), n 1 form a basis in the space L 2 [0,R]. 17

18 Calculating the Fourier coefficients The Fourier coefficients b l mn(s) are b0n(s) l = c NH2 H+1/2 R H J ν+1 (j ν,n )jν,n H+1 ( ) G 2,1 4R 2 1 (H ν)/2,n/2,1,1 (H + 1 ν)/2 4,2 s 2 jν,n 2. 0,1 H bmn(s) l = c NH2 m H+1/2 s 2H m R m H J ν+1 (j ν,n )jν,n m H+1 ( G 1,1 4R 2 3,1 s 2 jν,n 2 1 m H+1+ν,N/2 + H,1 m H+1 ν ) 2 2,m

19 Simplifying the Fourier coefficients The order of the Meijer G- on the previous slide can be reduced by 2 if we choose the following values of ν = ν(m): of simplification is: ν(m) = m 1 H. b l mn(s) = 2H+1 π (N 2)/2 Γ(N/2 + H)Γ(H + 1)sin(πH)R H Γ(N/2)J m 1 H+1 (j m 1 H,n )j H+1 m 1 H,n [2 (N 2)/2 Γ(N/2) J m+(n 2)/2(R 1 j m 1 H,n s) (R 1 j m 1 H,n s) (N 2)/2 δ m 0 ]. This result was proved by Malyarenko (2008b) for the case of R = 1. 19

20 A. Malyarenko (2008a), An optimal series expansion of the, J. Theor. Probab., 21, A. Malyarenko (2008b),, arxiv: v1 [math.pr] M. Yadrenko (1983), Spectral theory of random fields, Optimization Software, New York. 20