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Register a new userHigh order chaotic limits of wavelet scalograms under long--range dependence
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Let $G$ be a non--linear function of a Gaussian process ${X_t}_{tinmathbb{Z}}$ with long--range dependence. The resulting process ${G(X_t)}_{tinmathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with ${G(X_t)}_{tinmathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-It^o integral of order one or two. We show, however, that there are large classes of functions $G$ which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-It^o integral of order greater than two.
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A time-changed mixed fractional Brownian motion is an iterated process constructed as the superposition of mixed fractional Brownian motion and other process. In this paper we consider mixed fractional Brownian motion of parameters a, b and Hin(0, 1) time-changed by two processes, gamma and tempered stable subordinators. We present their main properties paying main attention to the long range dependence. We deduce that the fractional Brownian motion time-changed by gamma and tempered stable subordinators has long range dependence property for all Hin(0, 1).
Marcinkiewicz strong law of large numbers, ${n^{-frac1p}}sum_{k=1}^{n} (d_{k}- d)rightarrow 0 $ almost surely with $pin(1,2)$, are developed for products $d_k=prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} = sum_{l=-infty}^{infty}c_{k-l}^{(r)}xi_l^{(r)}$ are two-sided linear process with coefficients ${c_l^{(r)}}_{lin mathbb{Z}}$ and i.i.d. zero-mean innovations ${xi_l^{(r)}}_{lin mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|toinfty$, can be slow enough for ${x_k^{(r)}}$ to have long memory while ${d_k}$ can have heavy tails. The long-range dependence and heavy tails for ${d_k}$ are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The results provide a means to estimate how much (if any) long-range dependence and heavy tails a sequential data set possesses, which is done for real financial data. All of the stocks we considered had some degree of heavy tails. The majority also had long-range dependence. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.
For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
We consider the problem of recovering a $k$-sparse signal ${mbox{$beta$}}_0inmathbb{R}^p$ from noisy observations $bf y={bf X}mbox{$beta$}_0+{bf w}inmathbb{R}^n$. One of the most popular approaches is the $l_1$-regularized least squares, also known as LASSO. We analyze the mean square error of LASSO in the case of random designs in which each row of ${bf X}$ is drawn from distribution $N(0,{mbox{$Sigma$}})$ with general ${mbox{$Sigma$}}$. We first derive the asymptotic risk of LASSO in the limit of $n,prightarrowinfty$ with $n/prightarrowdelta$. We then examine conditions on $n$, $p$, and $k$ for LASSO to exactly reconstruct ${mbox{$beta$}}_0$ in the noiseless case ${bf w}=0$. A phase boundary $delta_c=delta(epsilon)$ is precisely established in the phase space defined by $0ledelta,epsilonle 1$, where $epsilon=k/p$. Above this boundary, LASSO perfectly recovers ${mbox{$beta$}}_0$ with high probability. Below this boundary, LASSO fails to recover $mbox{$beta$}_0$ with high probability. While the values of the non-zero elements of ${mbox{$beta$}}_0$ do not have any effect on the phase transition curve, our analysis shows that $delta_c$ does depend on the signed pattern of the nonzero values of $mbox{$beta$}_0$ for general ${mbox{$Sigma$}} e{bf I}_p$. This is in sharp contrast to the previous phase transition results derived in i.i.d. case with $mbox{$Sigma$}={bf I}_p$ where $delta_c$ is completely determined by $epsilon$ regardless of the distribution of $mbox{$beta$}_0$. Underlying our formalism is a recently developed efficient algorithm called approximate message passing (AMP) algorithm. We generalize the state evolution of AMP from i.i.d. case to general case with ${mbox{$Sigma$}} e{bf I}_p$. Extensive computational experiments confirm that our theoretical predictions are consistent with simulation results on moderate size system.
In this paper, we present a new Marshall-Olkin exponential shock model. The new construction method gives the proposed model further ability to allocate the common joint shock on each of the components, making it suitable for application in fields like reliability and credit risk. The given model has a singular part and supports both positive and negative dependence structure. Main dependence properties of the model is given and an analysis of stress-strength is presented. After a performance analysis on the estimator of parameters, a real data is studied. Finally, we give the multivariate version of the proposed model and its main properties.
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