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[MATH] Define [MATH] , where the minimum is taken over [MATH] and [MATH] . Cover [MATH] with [MATH] [MATH] -balls. Assumption about [MATH] (5): In ( 8.1 ), the [MATH] |
term is upper bounded by [MATH] . Here, we are effectively assuming a bound on the third derivative of [MATH] with respect to [MATH] over the compact sets [MATH] and [MATH] |
If [MATH] for some [MATH] then we must have [MATH] for some [MATH] that is at the center of one of the [MATH] -balls covering [MATH] . The nonlinear transfer of volume lemma implies that the probability of [MATH] for some [MATH] |
relative to [MATH] is upper bounded by [EQUATION] If [MATH] , this probability goes to zero as [MATH] Therefore, every [MATH] is a Lebesgue point of [MATH] , proving the lemma. |
We are now prepared to state and prove the immersivity theorem. Theorem 21 Suppose [MATH] and [MATH] satisfy the assumptions laid down in section 6 and suppose [MATH] . The delay map [MATH] |
is then immersive at every point of [MATH] with probability [MATH] relative to the ball [MATH] Proof. The proof follows by taking [MATH] through a countable sequence in the previous Lemma 20 and using the assumption made in section 6 about immersivity at fixed points. |
Discussion The delay map may be viewed in light of the Whitney embedding theorem . However, it has some characteristics of its own. One of these is the possibility that orbits of two distinct points can overlap. There are other distinctive characteristics related to periodic orbits and eigenvectors. |
In this article, we showed how to prove that the delay map is an embedding using the concept of Lebesgue points. For the delay map [MATH] |
with [MATH] to be an embedding with probability [MATH] relative to the ball [MATH] , we require the embedding dimension to satisfy [MATH] |
We conjecture that the delay mapping is an embedding for [MATH] The more restrictive [MATH] requirement comes in when applying the nonlinear transfer of volume lemma. The extra dimensions are used to absorb the effect of the nonlinear term. Some evidence for this conjecture may be found in our earlier work |
In our opinion, it would be desirable to obtain prevalence versions of classical theorems such as the Kupka-Smale theorem The differential topology proofs rely heavily on the bump function and genericity is weaker than almost sureness in probability. It is hoped that the technique based on Lebesgue points introduced he... |
# Source: arxiv 1806.07604 # Title: Multifractal characteristics and return predictability in the Chinese stock markets # Sections: all # Downloaded: 2026-03-03T04:47:35.655675+00:00 |
Multifractal characteristics and return predictability in the Chinese stock markets Abstract By adopting Multifractal detrended fluctuation (MF-DFA) analysis methods, the multifractal nature is revealed in the high-frequency data of two typical indexes, the Shanghai Stock Exchange Composite 180 Index (SH180) and the Sh... |
keywords: Multifractal Characteristics , Multifractal Detrended Fluctuation Analysis , Return Predictability Introduction As we know, market returns do not have auto-correlations, but exhibit nonlinear long memory behaviors, which corresponding to the multifractal nature (Muzy et al., 2001 ; Calvet and Fisher, 2002 ; J... |
As high market risks are accompanied with high returns, we could infer that strong multifractality in price dynamics will have high returns. However, such inferences are still lack of empirical evidence. |
The research framework of return predictability provides an avenue to uncover whether the multifractal characteristics can be employed as predictors to forecast returns. The return predictability has been received considerable research interests, because it can highlight the understanding of asset pricing in the academ... |
Inspired by the potential connections between multifractal nature and market volatility, it would be interesting to test whether the multifractal characteristic could be a return predictor. Our work will fill this gap. This paper is organized as follows: Data and methods are given in Sec. . Sec. presents the results of... |
Data and Methods 2.1 Data sets Our data, including the Shanghai Stock Exchange Composite 180 Index (SH180) and the Shenzhen Stock Exchange Composite Index (SZCI) in the Chinese stock markets, are retrieved from the finance database of Resset (). Both indexes cover a period from February 14, 2003 to December 31, 2015 in... |
[EQUATION] We regard the last price on each trading day as the closing price [MATH] on that day and the daily return [MATH] is defined in the following, |
[EQUATION] 2.2 Multifractal detrended fluctuation analysis (MF-DFA) For a given window of minutely returns [MATH] , we can define [MATH] as follows, |
[EQUATION] The series [MATH] is covered by [MATH] disjoint boxes and each box has the same size [MATH] . For our convenience, we label the sub-series in each box as, |
[EQUATION] In some cases, the whole series [MATH] cannot be exactly covered by [MATH] boxes, which means that we have to neglect some data points at the end of the series. In order to avoid this situation, we can utilize [MATH] boxes to cover the series, where [MATH] boxes cover from the beginning and [MATH] boxes cove... |
[EQUATION] where [MATH] can take any real value except for [MATH] . While [MATH] we have [EQUATION] according to l’Hôpital’s rule. By varying the value of [MATH] in the range from [MATH] to [MATH] , one can expect the detrended fluctuation function [MATH] scales with the size |
[MATH] , which reads [EQUATION] where [MATH] is the generalized Hurst index. Note that while [MATH] [MATH] is nothing but Hurst index [MATH] . The scaling exponents |
[MATH] , which is used to reveal the multifractality in the standard multifractal formalism based on partition function, can be obtained from the following traditional function for each [MATH] |
[EQUATION] where [MATH] is the fractal dimension of the geometric support of the multifractal measure (in our case [MATH] ). The local singularity exponent [MATH] of the measure [MATH] and its spectrum [MATH] |
are related to [MATH] through the Legendre transformation (Halsey et al., 1986 [EQUATION] Taking into account the statistical significance of the estimation of overall fluctuation functions, we focus on [MATH] |
We further employ these three parameters ( [MATH] [MATH] , and [MATH] ) to capture the overall characteristics of the multifractal spectrum. Parameter [MATH] stands for the width of multifractal spectrum, defined as [MATH] |
[MATH] quantitatively describes the dispersion of singularity exponents [MATH] , and thus measures the degree of heterogeneity for the probability measure of subsets on the overall fractal structure according to the definition of [MATH] |
(Zhou, 2007 . In practice, [MATH] is widely used to gauge the degree of multifractality (Jiang and Zhou, 2008b ; Zhou, 2009 . The larger the value of [MATH] is, the stronger the multifractal nature is. Parameter [MATH] is estimated via [MATH] , depicting the difference between the proportion of the subset with the mini... |
(Shimizu et al., 2002 ; Munõz-Diosdado and Río-Correa, 2006 [MATH] indicates the asymmetry of the spectrum. If the absolute value of [MATH] approaches to 0, the spectrum curve is more favorable to being symmetric. [MATH] means that the spectrum curve is right-hooked, indicating that the data set is dominated by the sub... |
The shape of the multifractal spectrum and the definition of [MATH] and [MATH] underlie a positive correlation between [MATH] and [MATH] . When [MATH] (respectively, [MATH] ), the multifractal spectrum is left-hooked (respectively, right-hooked), which means [MATH] (respectively, [MATH] ), and then we have [MATH] (resp... |
Empirical multifractal characteristics Using a moving window with a size of 5 days, we perform the multifractal analysis on the returns in each window by means of the MFDFA method. To have an impression that the multifractal spectrum is able to quantitatively capture the market dynamics, we present the results of multi... |
We also estimate the three characteristic parameters [MATH] of multifractal spectrum in the three windows and obtain [MATH] [MATH] , and [MATH] for window 1, [MATH] [MATH] , and [MATH] for window 2, [MATH] [MATH] , and [MATH] for window 3, respectively. For the width of multifractal spectra, we have [MATH] . This resul... |
By performing the multifractal analysis on the return series in each moving window, we will accumulate three series of the multifractal characteristics ( [MATH] [MATH] , and [MATH] ). Table lists the basic information of the cumulative return ( [MATH] ) and the multifractal characteristics ( [MATH] [MATH] , and [MATH] ... |
In Panel B of Table , one can find that there is no correlation between the daily returns following moving windows [MATH] and the three multifractal parameters ( [MATH] [MATH] , and [MATH] ), as their correlation coefficients are very close to 0 and none of them is significant. The cumulative returns in moving windows ... |
In Panel C of Table , we find that [MATH] and [MATH] exhibit very strong autocorrelated behaviors, since their autocorrelation coefficients of lags 1 and 5 are positive, large, and significant. The autocorrelations of cumulative returns [MATH] in moving windows are around 0.78 at lag 1 for both indexes and quickly fall... |
We report the results of augmented Dickey-Fuller (ADF) unit root tests and ARCH tests in Panels D and E of Table . For the ADF unit root test, the optimal lag length is determined according to the Schwarz information criterion. For both indexes, all the ADF statistics show the rejection of the null hypothesis of a unit... |
Predictive power of multifractal characteristics 4.1 In-sample tests The daily excess return [MATH] on day [MATH] is defined as the difference between the index return [MATH] and the market risk-free return [MATH] |
[EQUATION] And the corresponding multifractal characteristics on day [MATH] , denote as [MATH] [MATH] , and [MATH] , are estimated from day [MATH] to day [MATH] . We first separate the multifractal characteristic [MATH] into six groups according to whether they fall into the following six bins, [MATH] [MATH] [MATH] [MA... |
We further conduct the Granger causality tests between the excess returns [MATH] and the multifractal characteristics [MATH] [MATH] , and [MATH] . The corresponding null hypothesis of [MATH] is that [MATH] does not Granger cause [MATH] . The results are listed in Table . One can see that the excess return [MATH] is the... |
A standard univariate predictive regression framework is employed to test the predictive power of mulitfractal characteristics [MATH] from day [MATH] to day [MATH] on the excess returns on day [MATH] |
[EQUATION] where [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] . The corresponding results of in-sample tests are shown in Table for both indexes, in which the regression slopes, intercepts, and the adjusted [MATH] statistics are reported. The [MATH] -values, which are obtained from the NW [MATH] -tests (Newey and We... |
In panel A, we list the results of in-sample tests for the excess returns of SH180 index. Except [MATH] , all the other multifractal characteristics give negative regression slopes. However, none of them is statistically significant. We can observe that only the regression slope of [MATH] is significant at the level of... |
4.2 Comparison with the Chinese market volatility measures To compare the predictive power of multifractal characteristics with the Chinese market volatility measures, we perform the following regression, |
[EQUATION] where [MATH] is the realized market volatility from day [MATH] to [MATH] , which is estimated by summing the square of minutely returns in sample periods. |
Table. reports the results of the regression to Eq. ( 12 ). We find that the results in Table. are in accordance with those in Table. and that only the [MATH] of [MATH] is statistically significant for both indexes and the other predictors including the realized volatility are economically insignificant, which reveals ... |
4.3 Out-of-sample tests Out-of-sample tests are usually encouraged to evaluate the return predictability by excluding the using of future information and over-fitting in-sample tests. Two statistics, the [MATH] statistic (Campbell and Thompson, 2008 and adjusted MSFE statistic (Clark and West, 2007 , are employed to as... |
[EQUATION] where [MATH] is the actual excess return, [MATH] is the predicting excess return, and [MATH] is the benchmark of historical average returns. [MATH] measures the percent reduction in mean square forecast error for the predictive regression forecast relative to the historical average benchmark forecast. From t... |
Our out-of-sample tests are conducted in both expanding windows and moving windows. The detailed procedure is listed as follows. The predictive regression (Eq. ( 11 )) is estimated in each window and the obtained parameters are then used to generate the out-of-sample prediction for the day following that window. These ... |
Conclusion In this paper, we apply MF-DFA to detect the multifractal characteristics in high-frequency data of SH180 Index and SZSE Index. We find that both indexes exhibit strong multifractality. We propose three volatility measures based on the multifractal spectra ( [MATH] [MATH] , and [MATH] ) obtained from returns... |
One possible explanation of such predicting ability is that multifractal characteristics are considered as measures of market volatility and can be linked to market risk, the return predictability may be economically explained by the theory of risk premium (Merton, 1980 ; French et al., 1987 . Another possible explanat... |
Acknowledgments: This work was partially supported by the National Natural Science Foundation of China (71571121), the Shanghai Philosophy and Social Science Fund Project (2017BJB006) and the Fundamental Research Funds for the Central Universities (222201718006). |
# Source: arxiv 1806.07768 # Title: Anchoring and Binning the Coordinate Bethe Ansatz # Sections: all # Downloaded: 2026-03-02T08:54:16.146677+00:00 |
Anchoring and Binning the Coordinate Bethe Ansatz Jarah Evslin 1,2 1) Institute of Modern Physics, NanChangLu 509, Lanzhou 730000, China |
2) University of the Chinese Academy of Sciences, YuQuanLu 19A, Beijing 100049, China Abstract The Coordinate Bethe Ansatz (CBA) expresses, as a sum over permutations, the matrix element of an XXX Heisenberg spin chain Hamiltonian eigenstate with a state with fixed spins. These matrix elements comprise the wave functio... |
Introduction 1.1 Motivation Man has always sought to understand the origin of the Yang-Mills mass gap. In the instantaneous frame, it is a consequence of the ground state. This ground state may be realized, in the Schrodinger picture, as a wave functional which satisfies the Schrodinger equation |
. Despite decades of efforts, no such solution appears to be forthcoming. On the other hand, Yang-Mills theory in 3+1 dimensions is quite similar to the [MATH] nonlinear sigma model in [MATH] dimensions. Here also fractional instantons are somehow involved in the generation of a mass gap |
. Knowledge of the ground state and first excited state wave functionals of this model would unlock exciting doors, allowing a concrete understating of how the instantons generate the mass gap in the Minkowski theory, perhaps as a kind of infinite-dimensional generalization of the familiar story in quantum mechanics wi... |
Our motivation is based on an analogy, summarized in Table , between (i) The double well model in quantum mechanics, (ii) The [MATH] nonlinear sigma model and (iii) Yang-Mills theory. Consider the following states: (i) A position eigenstate corresponding to the point [MATH] , (ii) A wave functional which vanishes on al... |
In quantum mechanics, the mass gap may be seen as a consequence of a discrete choice in how the wave functions are connected across the barriers. Is there a similar story in quantum field theory? Does this cross-barrier bridge also render a monopole-operator tachyonic in Yang-Mills? After all, it is known that in some ... |
. To answer these questions, we need at least to understand the basic features of the ground state and first excited wave functionals. For example, does the first excited state wave functional have a node at the maximum of this potential, corresponding to [MATH] in the sigma model or a half-integral [MATH] in Yang-Mill... |
, manifested in the vacuum state? This sigma model is not only solvable but has already been solved . So, what are the wave functionals? A map between the [MATH] sigma model and the XXX Heisenberg spin chain was shown in |
at the level of low energy fluctuations and in in the full quantum theory. The former applies to a spin chain of any spin [MATH] , with strong coupling at small spin while the latter, reviewed in Appendix A , strictly speaking yields an equivalence only at infinite [MATH] , although finite [MATH] can be used as a defin... |
So what are the sigma model wave functionals? To actually take these spin chain solutions and map them to something intelligible on the sigma model side was Faddeev’s challenge to his students in |
. The map is known in the coordinate basis of spins in the spin chain, and so to meet the challenge one needs the matrix elements of the spin chain Hamiltonian eigenvectors with the coordinate states, which have definite spins at each lattice site. The challenge is indeed a challenge because, while many forms are by no... |
, each grows in complexity either with the length [MATH] of the chain or else with the distance of a coordinate state from a preferred spin state, such as the classical ground state. |
Our goal is to present a method for approximating the matrix elements which depends on the complexity of the state, but not directly on [MATH] . The individual lattice sites are replaced by bins. The intuition is that the states which survive to the continuum limit are those which are essentially homogeneous inside of ... |
1.2 Outline After a review of the XXX spin chain in Sec. , we begin in Sec. with the first key ingredient in our construction, the anchor. The CBA gives the matrix element [MATH] between a given spin chain basis state and a given energy eigenstate as a sum of phases [MATH] , one for each element [MATH] of the permutati... |
[EQUATION] In other words it is given by the Fourier transform of the density function [MATH] If [MATH] were a Gaussian distribution, this transform would be trivial. If it were close to a Gaussian distribution, one could perform the Fourier transform perturbatively, using a moment expansion of [MATH] . Unfortunately, ... |
The anchor is a permutation-dependent integral multiple of [MATH] which we we will subtract from the arguments [MATH] of the phases. We refer to the difference as the anchored argument [MATH] . Clearly subtracting the anchor does not affect the matrix elements, as these depend only upon [MATH] . Our first main result i... |
[EQUATION] is of order [MATH] . Our second main result, which is shown analytically using the binning approximation described below, is that the standard deviation of the anchored [MATH] is only of order [MATH] . The anchored argument [MATH] therefore provides a more convenient starting place for a perturbative calcula... |
The other key ingredient is introduced in Sec. . To calculate the moments of the density [MATH] , the elements of the group [MATH] are realized as one to one maps from the integers [MATH] to themselves. We divide this interval into [MATH] bins. For each permutation [MATH] one can determine how many elements of the [MAT... |
[EQUATION] where [MATH] is the number of elements [MATH] such that [MATH] and [MATH] is equal to [MATH] where [MATH] is a particular permutation such that [MATH] . Our third main result is our formula for [MATH] , or stated differently [MATH] as a function of the [MATH] , in Eqs. ( 4.9 ) and ( 4.23 ). The moments of [M... |
Finally in Sec. we will, in the case of the matrix element between the classical and quantum ground states, apply the techniques introduced above to calculate the [MATH] contribution to the second moment of the anchored [MATH] . We will see explicitly that its coefficient is small, but it does not vanish. |
The Antiferromagnetic XXX Heisenberg Spin Chain The [MATH] sigma model is the continuum limit of a spin chain with an infinite spin at each lattice site. Classically, the spin squared corresponds to the inverse coupling |
and so low spin corresponds to a high coupling. In particular, at low spin one describes the sigma model at strong coupling and one does not expect a sensible description of individual instantons. Therefore, it will be essential for us to eventually extend our analysis to higher spin. However, in the present note we wi... |
2.1 Finite Chain The spin [MATH] Heisenberg spin chain consists of [MATH] lattice sites. At each lattice site lies a Hilbert space [MATH] with basis [MATH] . The total Hilbert space is the [MATH] -fold tensor product of these [MATH] . At each lattice site [MATH] lies an [MATH] Lie algebra with generators [MATH] satisfy... |
[EQUATION] This algebra acts on the [MATH] Hilbert space at the site [MATH] , according to the usual 2-dimensional representation such that |
[EQUATION] The XXX spin chain corresponds to the Hamiltonian [EQUATION] where [MATH] is the identity. We will let the constant [MATH] be positive, corresponding to the antiferromagnetic spin chain. Although the eigenvalues of [MATH] depend on [MATH] , in this note we will only be interested in the eigenvectors, which a... |
Any state can be decomposed into the basis consisting of the tensor product of the [MATH] bases at each lattice site. An element of the basis is a string of [MATH] ’s and [MATH] ’s. It is described by the set of positions [MATH] of the [MATH] th [MATH] for all [MATH] . Therefore an arbitrary state [MATH] is fully chara... |
[EQUATION] The Hamiltonian commutes with rigid rotations, which are [MATH] Lie algebra with basis [EQUATION] Therefore it can be diagonalized simultaneously with [MATH] . As a result, each Hamiltonian eigenstate can be taken to have a definite number [MATH] of spin downs. |
For all Hamiltonian eigenstates [MATH] , the elements [MATH] are given by the coordinate Bethe Ansatz [EQUATION] where [MATH] is the permutation corresponding to [MATH] . The information about the state is contained in the functions [MATH] and [MATH] which are related by |
[EQUATION] and by the Bethe equation [EQUATION] where [MATH] is an integer. In fact, a state is characterized by just the set of [MATH] The ground state for example corresponds to |
[EQUATION] The right hand side of Eq. ( 2.1 ) contains an [MATH] , which we have set to unity. However in the classical limit it is instead set to zero, in which case the lowest energy state of [MATH] becomes a classical ground state, such as |
[EQUATION] which corresponds to [EQUATION] In most of this paper we will restrict our attention to the matrix element between the classical ground state [MATH] and the quantum ground state [MATH] |
[EQUATION] The generalization of our results to other matrix elements with well-behaved continuum limits is essential for our goals. While we suspect that this will be a straightforward generalization of the calculations below, we leave these extension to future work. |
2.2 The Thermodynamic Limit One may automatically solve Eq. ( 2.7 ) by introducing spectral parameters [MATH] , related to [MATH] and [MATH] by |
[EQUATION] so that [EQUATION] and [EQUATION] We recall that [MATH] and [MATH] and so in Eqs. ( 2.14 ) and ( 2.15 ) the ranges of both ArcTan and ArcCot must be taken to be [MATH] . Using ( 2.13 ) Bethe’s equation ( 2.8 ) can be rewritten as a condition on the spectral parameters |
[EQUATION] It will prove more convenient to rewrite Bethe’s equation using ( 2.14 ) and ( 2.15 ) to obtain [EQUATION] We have kept ArcCot [MATH] and ArcTan [MATH] . We would like to replace the [MATH] ArcCot above with [MATH] , where ArcTan [MATH] . However, using our conventions [MATH] ArcCot [MATH] . Therefore, to co... |
[EQUATION] To pass to the continuum limit, one replaces the lattice site index [MATH] with [EQUATION] Sometimes it is convenient to replace [MATH] by [MATH] in this expression to make it symmetric in [MATH] , however this will only affect subdominant contributions in [MATH] and will not affect our main results here. No... |
[EQUATION] By abuse of notation, we will drop the tildes and write simply [MATH] for both the original discrete function and its continuous interpolation. The interpolation is not uniquely defined, however if one imposes ( 2.18 ) then the choice of interpolation is irrelevant, since the equation only restricts the valu... |
To fix [MATH] at all [MATH] , one replaces the sum in Eq. ( 2.18 ) with an integral [EQUATION] so that the spectral function [MATH] is determined by |
[EQUATION] The replacement ( 2.21 ) is not an equality. It changes the equation. The solutions [MATH] will not be solutions of the original equation, even at the lattice sites [MATH] . It is expected that this correction is subdominant in the [MATH] expansion. However, these subleading corrections to the [MATH] may in ... |
2.3 The Ground State One can now solve ( 2.22 ) to find the above functions of [MATH] for the quantum ground state [MATH] . First, let us define the density |
[EQUATION] which is unrelated to the density of phases [MATH] introduced above. The derivative of Eq. ( 2.22 ) with respect to [MATH] is |
[EQUATION] Now multiply through by [MATH] . The function [MATH] is a bijection and so we can pull back any function [MATH] to obtain [MATH] . Let [MATH] and [MATH] . This allows us to rewrite the entire equation using functions of [MATH] and [MATH] |
[EQUATION] where the integration measure was converted using [EQUATION] The equation ( 2.25 ) is usually solved using Fourier transforms. We will review the argument here, as we need to go a few steps beyond the textbook treatment to obtain all functions of [MATH] explicitly. The Fourier transform of the left hand side... |
[EQUATION] The integrand has simple poles at [MATH] . If [MATH] [MATH] ) then the integrand vanishes exponentially for a large semicircular contour on the upper (lower) half of the complex plane. The corresponding contour encircles the pole at [MATH] [MATH] ), where the residue is [MATH] [MATH] ). The contour is counte... |
[EQUATION] Defining the Fourier transform of the density by [EQUATION] the Fourier transform allows Eq. ( 2.25 ) to be rewritten |
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