flicker noise and the estimation of the allan variance flicker noise is a random process observed in a variety of contexts including current fluctuations in metal film and semiconductor devices loudness fluctuations in speech and music and neurological patterns the quadratic mean convergence of appropriate estimates of the allan variance for flicker noise is established when the latter is modeled as a stochastic process with stationary increments a precise asymptotic expression of the mean square error is given along with the rate of convergence

a wavelet based kl like expansion for wide sense stationary random processes we describe a wavelet based series expansion for wide sense stationary processes the expansion coefficients are uncorrelated random variables a property similar to that of a karhunen loeve kl expansion unlike the kl expansion however the wavelet based expansion does not require the solution of the eigen equation and does not require that the process be time limited this expansion also has advantages over fourier series which is often used as an approximation to the kl expansion in that it completely eliminates correlation and that the computation for its coefficients are more stable over large time intervals the basis functions of this expansion can be obtained easily from wavelets of the lemaire meyer 1990 type and the power spectral density of the process finally the expansion can be extended to some nonstationary processes such as those with wide sense stationary increments

on the quasilinear elliptic problems with critical sobolev hardy exponents and hardy terms abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we study the critical quasilinear elliptic problems p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p p q p s p n s n p p t p n t n p p s and p t are the critical sobolev hardy exponents via variational methods we deal with the conditions that ensure the existence of positive solutions for the equation the results depend crucially on the parameters p q s and

dimension estimation of discrete time fractional brownian motion with applications to image texture classification fractional brownian motion fbm is a suitable description model for a large number of natural shapes and phenomena in applications it is imperative to estimate the fractal dimension from sampled data namely discrete time fbm dfbm to this aim the increment of dfbm referred to as discrete time fractional gaussian noise dfgn is invoked as an auxiliary tool the regular part of dfgn is first filtered out via levinson s algorithm the power spectral density of the regular process is found to satisfy a power law that its exponent can be well fitted by a quadratic function of fractal dimension a new method is then proposed to estimate the fractal dimension of dfbm from the given data set the computational complexity and statistical properties are investigated moreover the proposed algorithm is robust with respect to amplitude scaling and shifting as well as time shifting on the data finally the effectiveness of this estimator is demonstrated via a classification problem of natural texture images

multiresolution analysis of a class of nonstationary processes processing nonstationary signals is an important and challenging problem we focus on the class of nonstationary processes with stationary increments of an arbitrary order and place them in a multiscale framework unlike other related studies we concentrate on the discrete time analysis and derive a number of new results in addition to placing the related existing ones in the same framework we extend the study to various parametric models for which we derive the resulting multiresolution description we show that wide sense stationarity may be achieved by adequately selecting the analysis wavelet after generalizing the study to wavelet packet analysis we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

invariances laplacian like wavelet bases and the whitening of fractal processes in this contribution we study the notion of affine invariance specifically invariance to the shifting scaling and rotation of the coordinate system as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional brownian motion fbm fields in particular using a rigorous and powerful distribution theoretic formulation we extend previous results of blu and unser 2006 to the multivariate case showing that polyharmonic splines and fbm processes can be seen as the deterministic vs stochastic solutions to an identical fractional partial differential equation that involves a fractional laplacian operator we then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional laplacian which also turns out to be the whitening operator for fbm fields this fact allows us to study the probabilistic properties of the wavelet transform coefficients of fbm like processes leading for instance to ways of estimating the hurst exponent of a multiparameter process from its wavelet transform coefficients we provide theoretical and experimental verification of these results to complement the toolbox available for multiresolution processing of stochastic fractals we also introduce an extended family of multidimensional multiresolution spaces for a large class of separable and nonseparable lattices of arbitrary dimensionality

multiple positive solutions for semilinear elliptic equations involving multi singular inverse square potentials and concave convex nonlinearities abstract in this paper we deal with the existence and multiplicity of positive solutions for the multi singular semilinear elliptic equations u i 1 k i x a i 2 u u 2 2 u u q 2 u x where r n n 3 is a smooth bounded domain such that the points a i i 1 2 k k 2 are different 0 i n 2 2 2 0 1 q 2 and 2 2 n n 2 the results depend crucially on the parameters q and i for i 1 2 k

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

low rank variance approximation in gmrf models single and multiscale approaches we present a versatile framework for tractable computation of approximate variances in large scale gaussian markov random field estimation problems in addition to its efficiency and simplicity it also provides accuracy guarantees our approach relies on the construction of a certain low rank aliasing matrix with respect to the markov graph of the model we first construct this matrix for single scale models with short range correlations and then introduce spliced wavelets and propose a construction for the long range correlation case and also for multiscale models we describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse irregular and noisy measurements and to a gravity inversion problem

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

on rate distortion models for natural images and wavelet coding performance operational rate distortion rd functions of most natural images when compressed with state of the art wavelet coders exhibit a power law behavior dpropr gamma at moderately high rates with gamma being a constant depending on the input image deviating from the well known exponential form of the rd function dprop2 xir for bandlimited stationary processes this paper explains this intriguing observation by investigating theoretical and operational rd behavior of natural images we take as our source model the fractional brownian motion fbm which is often used to model nonstationary behaviors in natural images we first establish that the theoretical rd function of the fbm process both in 1 d and 2 d indeed follows a power law then we derive operational rd function of the fbm process when wavelet encoded based on water filling principle interestingly both the operational and theoretical rd functions behave as dpropr gamma for natural images the values of gamma are found to be distributed around 1 these results lend an information theoretical support to the merit of multiresolution wavelet compression of self similar processes and in particular natural images that can be modelled by such processes they may also prove useful in predicting performance of rd optimized image coders

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

an exact estimate result for a semilinear equation with critical exponent and prescribed singularity abstract we consider the singular boundary value problem u v x u h x u u 2 1 in u x 0 in u x 0 on where 0 n 2 2 2 0 1 2 2 n n 2 h x is a given function and v x has prescribed finitely many singular points our goal in this paper is to establish some existence and multiplicity results for above problem when 0 for some 0 and obtain exact estimate for extremal value 2 h x 0 for above problem

asymptotic decorrelation of between scale wavelet coefficients in recent years there has been much interest in the analysis of time series using a discrete wavelet transform dwt based upon a daubechies wavelet filter part of this interest has been sparked by the fact that the dwt approximately decorrelates certain stochastic processes including stationary fractionally differenced fd processes with long memory characteristics and certain nonstationary processes such as fractional brownian motion it is shown that as the width of the wavelet filter used to form the dwt increases the covariance between wavelet coefficients associated with different scales decreases to zero for a wide class of stochastic processes these processes are gaussian with a spectral density function sdf that is the product of the sdf for a not necessarily stationary fd process multiplied by any bounded function that can serve as an sdf on its own we demonstrate that this asymptotic theory provides a reasonable approximation to the between scale covariance properties of wavelet coefficients based upon filter widths in common use our main result is one important piece of an overall strategy for establishing asymptotic results for certain wavelet based statistics

analysis and synthesis of 1 f processes via shannon wavelets 1 f processes can be very useful in modeling processes with long term correlation we propose analysis and synthesis procedures to express these processes in terms of the shannon wavelet unlike previous techniques our analysis procedure generates uncorrelated decomposition coefficients for the 1 f process this is done by taking onto account and then removing the residual correlation between the wavelet components the analysis procedure is the major contribution of this work the proposed synthesis algorithm which is a byproduct of the proposed analysis algorithm is competitive with other techniques

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

wavelet packets of fractional brownian motion asymptotic analysis and spectrum estimation this paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional brownian motion it also discusses the convergence speed to the limit autocorrelation function when the input random process is either a fractional brownian motion or a wide sense stationary second order random process the analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the shannon paraunitary filters from this analysis we derive wavelet packet based spectrum estimation for fractional brownian motions and wide sense stationary random processes experimental tests show good results for estimating the spectrum of 1 f processes

wavelet entropy and fractional brownian motion time series abstract we study the functional link between the hurst parameter and the normalized total wavelet entropy when analyzing fractional brownian motion fbm time series these series are synthetically generated both quantifiers are mainly used to identify fractional brownian motion processes l zunino d g perez m garavaglia o a rosso characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform fractals 12 2 2004 223 233 the aim of this work is to understand the differences in the information obtained from them if any

nehari manifold and existence of positive solutions to a class of quasilinear problems abstract in this paper existence and multiplicity results to the following nonlinear elliptic equation p u u q 2 u u p 2 u u 0 in r n together with mixed dirichlet neumann or neumann boundary conditions are established here p u denotes the p laplacian operator 1 q p n p np n p and is a positive real parameter the study is based on the extraction of palais smale sequences in the nehari manifold

a nonlinear elliptic equation with critical exponent estimates for extremal values abstract in this work we study the values of p q the supremum of the set of related to existence and multiplicity of positive solutions of semilinear elliptic equations involving a critical sobolev exponent and establish uniform lower bounds for

some distributional properties of the continuous wavelet transform of random processes without finite moment conditions some properties of random processes such as stationarity and self similarity are characterized via corresponding properties of their wavelet transform anyone of these distributional properties of the wavelet transform characterizes the corresponding property of the increments of the random process of order equal to the order of regularity of the analyzing wavelet extensions of these results to random fields are then indicated

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

estimation of 1 f signals on the basis of curve fitting an algorithm is proposed for the identification of fractal signals with wavelet based models from corrupted data this algorithm is based on curve fitting and wavelet transformation it is proved that whenever spl gamma is greater than 0 the algorithm provides an almost consistent estimation moreover the estimated parameters are asymptotically gaussian distributed a mean square asymptotic convergence rate of the estimated parameters has also been established simulation results verify the efficiency of the proposed algorithm

a note on the discrete wavelet transform of second order processes some properties of second order random processes such as stationarity and scaling are characterized via corresponding multiscale properties of their discrete wavelet transform the single scale characterizations valid in the continuous setting do not extend to the discrete setting some extensions of these results are also indicated

characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform dynamic study abstract we analyze within the wavelet theory framework the wandering over a screen of the centroid of a laser beam after it has propagated through a time changing laboratory generated turbulence following a previous work fractals 12 2004 223 two quantifiers are used the hurst parameter h and the normalized total wavelet entropy the temporal evolution of both quantifiers obtained from the laser spot data stream is studied and compared this allows us to extract information on the stochastic process associated with the turbulence dynamics

on the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes in this correspondence we develop an asymptotic theory for the correlations of wavelet coefficients of the discrete wavelet transform dwt for fractionally differenced processes it provides a theoretical justification for the approximate decorrelation property of the dwt for fractionally differenced processes in addition it provides insights on how the length of the wavelet filter affects the within scale correlations and the between scale correlations differently for within scale correlations increasing the length of the wavelet filter increases the rate of decay as the two wavelet coefficients get further apart while for between scale correlations using a wavelet filter that is long enough can reduce the between scale correlations even for wavelet coefficients that are close together

a nonlinear elliptic pde with two sobolev hardy critical exponents in this paper we consider the following pde involving two sobolev hardy critical exponents label 0 1 left begin aligned delta u lambda frac u 2 s 1 1 x s 1 frac u 2 s 2 1 x s 2 0 quad rm in omega quad quad quad 0 1 u 0 quad rm on omega end aligned right where 0 s2 s1 2 0 ne lambda in mathbb r and 0 in partial omega the existence or nonexistence for least energy solutions has been extensively studied when s1 0 or s2 0 in this paper we prove that if 0 s2 s1 2 and the mean curvature of partial omega at 0 h 0 0 then 0 1 has a least energy solution therefore this paper has completed the study of 0 1 for the least energy solutions we also prove existence or nonexistence of positive entire solutions of 0 1 with omega mathbb r n under different situations of s1 s2 and

wavelet spectrum and its characterization property for random processes the wavelet spectrum of a random process comprises the variances of the wavelet coefficients of the process computed within each scale this paper investigates the possibility of using the wavelet spectrum obtained from a continuous wavelet transform cwt to uniquely represent the second order statistical properties of random processes particularly stationary processes and long memory nonstationary processes as is well known the fourier spectrum of a stationary process is mathematically equivalent to the autocovariance function acf and thus uniquely determines the second order statistics of the process this characterization property is shown to be possessed also by the wavelet spectrum under very mild regularity conditions that are easily satisfied by many widely used wavelets it is also shown that under suitable regularity conditions the characterization property remains valid for processes with stationary increments including 1 f noise

on the continuous wavelet transform of second order random processes some second order properties of random processes such as periodic correlation stationarity harmonizability self similarity are characterized via corresponding properties of their wavelet transform any one of these properties of the wavelet transform characterizes the corresponding property of the increments of the random process of order equal to the order of regularity of the analyzing wavelet these results are then specialized to fractional brownian motion and other self similar processes

a fast estimation algorithm on the hurst parameter of discrete time fractional brownian motion the purpose of this paper is to demonstrate that the discrete time fractional gaussian noise dfgn is a regular process based on this property a fast algorithm with low computational cost is proposed to estimate the hurst parameter h which is an important parameter in fractional brownian motion fbm this algorithm is robust under amplitude shift invariant to time shift and unaffected by a scaling factor in power spectral density psd finally the computational complexity is also investigated

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

an exact estimate result for a class of singular equations with critical exponents abstract we consider the singular boundary value problem u h x u u p 1 in u 0 on with p 2 n n 2 0 1 it is well known that there exists 0 such that the problem has a solution for all 0 and no solution for we obtain an exact result for p h

on the correlation structure of the wavelet coefficients of fractional brownian motion shows that the interdependence of the discrete wavelet coefficients of fractional brownian motion defined by normalized correlation decays exponentially fast across scales and hyperbolically fast along time

schur reduction of trees and extremal entries of the fiedler vector abstract we study the eigenvectors of laplacian matrices of trees the laplacian matrix is reduced to a tridiagonal matrix using the schur complement this preserves the eigenvectors and allows us to provide formulas for the ratio of eigenvector entries we also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and perron values the results are then applied to the fiedler vector here we locate the extremal entries of the fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path

wavelet packets of nonstationary random processes contributing factors for stationarity and decorrelation the paper addresses the analysis and interpretation of second order random processes by using the wavelet packet transform it is shown that statistical properties of the wavelet packet coefficients are specific to the filtering sequences characterizing wavelet packet paths these statistical properties also depend on the wavelet order and the form of the cumulants of the input random process the analysis performed points out the wavelet packet paths for which stationarization decorrelation and higher order dependency reduction are effective among the coefficients associated with these paths this analysis also highlights the presence of singular wavelet packet paths the paths such that stationarization does not occur and those for which dependency reduction is not expected through successive decompositions the focus of the paper is on understanding the role played by the parameters that govern stationarization and dependency reduction in the wavelet packet domain this is addressed with respect to semi analytical cumulant expansions for modeling different types of nonstatonarity and correlation structures the characterization obtained eases the interpretation of random signals and time series with respect to the statistical properties of their coefficients on the different wavelet packet paths

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

stochastic transforms for jump diffusion processes combined with related backward stochastic differential equations abstract this study considers three stochastic transforms for jump diffusion processes the stochastic laplace transform stochastic fourier transform and stochastic wavelet transform first we introduce the stochastic laplace transform for processes adapted by brownian filtration as a solution for complex number valued backward stochastic differential equations bsdes this transform can be explained well by the girsanov theorem which also allows us to define the stochastic laplace transform for jump diffusion processes directly based on this perspective we give natural definitions of the stochastic fourier transform and stochastic wavelet transform the advantages of these stochastic transforms are all related to the uniqueness of the processes compared with the classical transforms the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes while they also agree with the corresponding parameters in the classical transforms which can represent the frequency property of the processes in addition these three stochastic transforms can also be regarded as the solutions of related bsdes with jumps

1 f noise decomposition in random telegraph signals using the wavelet transform abstract by using the continuous wavelet transform with haar basis the second order properties of the wavelet coefficients are derived for the random telegraph signal rts and for the 1 f noise which is obtained by summation of many rtss the correlation structure of the haar wavelet coefficients for these processes is found for the wavelet spectrum of the 1 f noise some characteristics related to the distribution of the relaxation times of the rts are derived a statistical test based on the characterization of the time evolution of the scalogram is developed which allows to detect non stationarity in the times s which compose the 1 f process and to identify the time scales of the relaxation times where the non stationarity is localized the proposed method allows to distinguish noise signals with 1 f power spectral density generated by rtss and thus gives informations on the origin of this type of 1 f noise which cannot be obtained using the fourier transform or other methods based on second order statistical analysis the reported treatment is applied to both simulated and experimental signals the present analysis is based on the mcwhorter 1 f noise and germanium surface properties in r h kingstone ed semiconductor surface physics university of pennsylvania press philadelphia pa 1957 pp 207 228 model of low frequency electric noise and the obtained results are expected to prove especially useful for semiconductor devices

statistical study of the wavelet analysis of fractional brownian motion we present a statistical study of wavelet coefficients of a fractional brownian motion a central limit theorem for empirical variances of exact wavelet coefficients is given under conditions on the mother wavelet and the choice of scales a limit theorem is given for fitted wavelet coefficients computed from a time series it provides an estimator for the self similarity parameter of gaussian time series

wavelet packet decompositions for the analysis of 2 d fields with stationary fractional increments nonstationary two dimensional random fields with wide sense stationary fractional increments are introduced using a fractional order for their increments in two directions of space some of their properties are studied and their value in modeling nonstationary possibly anisotropic fields is shown via examples a separable wavelet packet analysis proves to be a useful tool in analyzing this kind of texture as it leads to stationary wavelet packet coefficients

on the relationship between the hurst exponent the ratio of the mean square successive difference to the variance and the number of turning points abstract the long range dependence of the fractional brownian motion fbm fractional gaussian noise fgn and differentiated fgn dfgn is described by the hurst exponent h considering the realizations of these three processes as time series they might be described by their statistical features such as half of the ratio of the mean square successive difference to the variance a and the number of turning points t this paper investigates the relationships between a and h and between t and h it is found numerically that the formulae a h a e b h in case of fbm and a h a b h c for fgn and dfgn describe well the a h relationship when t h is considered no simple formula is found and it is empirically found that among polynomials the fourth and second order description applies best the most relevant finding is that when plotted in the space of a t the three process types form separate branches hence it is examined whether a and t may serve as hurst exponent indicators some real world data stock market indices sunspot numbers chaotic time series are analyzed for this purpose and it is found that the h s estimated using the h a relations expressed as inverted a h functions are consistent with the h s extracted with the well known wavelet approach this allows to efficiently estimate the hurst exponent based on fast and easy to compute a and t given that the process type fbm fgn or dfgn is correctly classified beforehand finally it is suggested that the a h relation for fgn and dfgn might be an exact shifted 3 2 power law

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated