3 See [MEY 90] for traditional results on operators

Scan UPC Symbol In .NET FrameworkUsing Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.

Self-similar Processes

Encode GTIN - 12 In .NET FrameworkUsing Barcode creation for VS .NET Control to generate, create UPC A image in .NET framework applications.

THEOREM 54 ([BEN 97]) Let X be a self-similar Gaussian (r, H) self-similar process, with r-stationary increments, with 0 < H < 1: X admits the following harmonic representation: X(x) = eix

Scan UPC Code In Visual Studio .NETUsing Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications.

Rd r 1 (ix )k k=0 k! d | | 2 +H S | |

Barcode Drawer In VS .NETUsing Barcode encoder for .NET framework Control to generate, create barcode image in .NET framework applications.

TF (W )(d )

Bar Code Decoder In VS .NETUsing Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications.

X is the unique solution of the following stochastic elliptic differential equation: LX = W Q(D)X(0) = 0 for any Q such that d Q < r As a particular case, we can mention the harmonic representation [MAN 68] of fractional Brownian motion of parameter H: r = 1 and S 1: BH (t) = eix 1

UPC A Maker In C#.NETUsing Barcode generation for .NET framework Control to generate, create UPC-A Supplement 2 image in Visual Studio .NET applications.

| | 2 +H

Create UPC A In VS .NETUsing Barcode maker for ASP.NET Control to generate, create GS1 - 12 image in ASP.NET applications.

TF (W )(d )

UPC-A Supplement 5 Maker In VB.NETUsing Barcode generation for .NET Control to generate, create UPC-A Supplement 5 image in VS .NET applications.

523 Hyperbolic processes DEFINITION 53 The operator L de ned in (108) is called hyperbolic if its symbol 1 d is of the form ( ) = i=1 | i |Hi + 2 THEOREM 55 (FRACTIONAL B ROWNIAN SHEET [LEG 99]) Fractional Brownian sheet, de ned as:

GS1-128 Drawer In Visual Studio .NETUsing Barcode creator for VS .NET Control to generate, create UCC-128 image in .NET framework applications.

X(x) =

Creating Bar Code In Visual Studio .NETUsing Barcode generation for .NET framework Control to generate, create barcode image in VS .NET applications.

eixi i 1 | i |Hi + 2

Generating UPCA In VS .NETUsing Barcode printer for .NET Control to generate, create UPC Code image in VS .NET applications.

Rd i=1

USPS OneCode Solution Barcode Printer In VS .NETUsing Barcode drawer for .NET Control to generate, create USPS Intelligent Mail image in .NET applications.

TF (W )(d )

Code39 Generator In Visual Studio .NETUsing Barcode generation for ASP.NET Control to generate, create Code-39 image in ASP.NET applications.

satis es the following equality: X( 1 x1 , , d xd ) =

Generating GTIN - 128 In JavaUsing Barcode generation for Java Control to generate, create GS1 128 image in Java applications.

i=1 L d

Generate UPCA In C#Using Barcode creator for .NET framework Control to generate, create UPC A image in VS .NET applications.

Hi X(x1 , , xd ) i

Barcode Generator In .NETUsing Barcode maker for ASP.NET Control to generate, create bar code image in ASP.NET applications.

COROLLARY 51 When 1 = = d and H = H1 + + Hd , the hyperbolic process X obtained is self-similar with parameter H, with H between 0 and d In contrast with the elliptic case, H > 1 is hence allowed, though the Brownian fractional sheet is non-derivable

Encode UCC - 12 In Visual Studio .NETUsing Barcode encoder for ASP.NET Control to generate, create GTIN - 12 image in ASP.NET applications.

Scaling, Fractals and Wavelets

EAN 13 Printer In JavaUsing Barcode generation for Java Control to generate, create EAN-13 Supplement 5 image in Java applications.

524 Parabolic processes Let A be a pseudo-differential operator of dimension n 1, ie, its symbol is a function of Rn 1 in R Let L be the pseudo-differential operator of dimension n, whose symbol is a function of R Rn 1 in R and de ned by L = t A Let us consider the stochastic differential equation LX = W By analogy with the classi cation of operators, X is said to be parabolic The most prominent example is the Ornstein-Uehlenbeck process:

Drawing Code 39 Extended In JavaUsing Barcode generator for Java Control to generate, create Code39 image in Java applications.

OU (t, x) =

DataMatrix Maker In VB.NETUsing Barcode printer for VS .NET Control to generate, create DataMatrix image in Visual Studio .NET applications.

e (t s)A W (ds, dxy)

The operator t is renormalized with a factor 1 ; the operator A can be 2 renormalized with an arbitrary factor Generally, a parabolic process is not self-similar 525 Wavelet decomposition In this section, we expand Gaussian self-similar processes on a wavelet basis, which hence also constitutes a basis for the self-reproducing Hilbert space of the process4 5251 Gaussian elliptic processes Let u , with u Ed , be a Lemari -Meyer generating system [MEY 90] Let , with d , be the generated orthonormal basis of wavelets Let us assume that X veri es hypotheses and notations, as in Theorem 54 Let us de ne u , with u Ed , with the harmonic representation: (x) =

eix

r 1 (ix )k k=0 k! d 2 +H S | | | |

TF ( u )(d )

Let us then de ne the associated family of wavelets , with d THEOREM 56 ([BEN 97]) There is a sequence of normalized Gaussian normal random 2D variables such that: X(x) =

d

2 j(r 1+H) (x)

If X is self-similar in the usual sense, then this decomposition is a renormalized distribution of mass, as de ned in section 513

4 See [NEV 68] for self-reproducing Hilbert spaces

Self-similar Processes

5252 Gaussian hyperbolic process THEOREM 57 With the same notations as those of Theorem 56, we obtain the decomposition: X(x) =

( 1 ,, d ) ( 1 )d

2 (n1 H1 + +nd Hd ) 1 (x1 ) d (xd ) 1 ,, d

where the sequence 1 ,, d consists of normalized Gaussian random 2D variables Hyperbolic processes enable us to model multiscale random structures, with preferred directions 526 Renormalization of sums of correlated random variable Let BH denote fractional Brownian motion of parameter H Let us consider the increments of size h > 0: Xk = h H (BH ((k + 1)h) BH (kh)) The following properties are well-known: X0 is a normalized Gaussian random variable; the sequence (Xk ) is stationary; a law of large numbers can be written as: 1 n + n + 1 lim