Iid Sequence Research Articles

Asymptotic Extremal Distribution for Non-Stationary, Strongly-Dependent Data

Fisher-Tippet-Gnedenko classical theory shows that the normalized maximum of n iid random variables with distribution F belonging to a very wide class of functions, converges in law to an extremal distribution H, that is determined by the tail of F. Extensions of this theory from the iid case to stationary and weak dependent sequences are well known from the work of Leadbetter, Lindgreen and Rootzén. In this paper, we present a very simple class of random processes that runs from iid sequences to non-stationary and strongly dependent processes, and we study the asymptotic behavior of its normalized maximum. More interesting, we show that when the process is strongly dependent, the asymptotic distribution is no longer an extremal one, but a mixture of extremal distributions. We present very simple theoretical and simulated examples of this result. This provides a simple framework to asymptotic approximations of extremes values not covered by classical extremal theory and its well-known extensions.

Finite-memory elephant random walk and the central limit theorem for additive functionals

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well-known result with a long history. In this paper, we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from Gut and Stadtmüeller (2018). We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

On the robust regression for a censored response data in the single functional index model

Let be an independent and identically distributed (iid) sequence of interest random variables (rv) distributed as T when the latter is subject to an independent censored rv C. In this work, we built and study a family of robust non parametric estimators for a regression function based on a kernel method. The covariates data are random functional type iid linked to single-index model. Under general conditions, we establish the pointwise almost sure convergence with rate an asymptotic normality of the estimator. The asymptotic standard deviation is explicitly given. By product, confidence bands are given and the special case is studied. Finally, simulations are drawn to illustrate both quality of fit and robustness.

Higher order asymptotics for large deviations – Part I

For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We show that the results are applicable to Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type.

Perturbation solution and welfare costs of business cycles in DSGE models

Lucas (1987) argues that the removal of cyclical fluctuations would barely improve economic welfare. He considers a risk-averse consumer valuing exogenously given streams of consumption Ct=Cegte−σ2/2+σϵt driven by an iid sequence of draws ϵt from a standard normal distribution. This allows him to change the size σ2 of the fluctuations without changing the mean of the process. In production economies, too, uncertainty is typically introduced in form of multiplicative, log-normally distributed shocks so that mean-preserving spreads can be analyzed in an analogous way. However, only few stochastic dynamic general equilibrium (DSGE) models admit an analytic solution. The most prevalent method to solve these models, perturbation methods, obtain an approximation to the stochastic model by perturbing the solution of the model’s deterministic counterpart with respect to the uncertainty parameter σ. Yet, widely available formulae to compute perturbation solutions are based only on the perturbation of σϵt and do not adequately capture the additional deviation −σ2/2 in the mean between the stochastic model and its deterministic version. We show within a model admitting an analytical solution that a second-order approximation of the welfare criterion also requires to perturb the mean. Thus, welfare measures based on the standard procedures for second-order solutions are (seriously) biased by a purely exogenous mean effect. We develop a general procedure of computing second-order accurate approximations of welfare gains or losses in the canonical DSGE model by extending the computation of second-order solutions pioneered by Schmitt-Grohé and Uribe (2004) to allow for mean preserving increases in the size of shocks. We apply our method to the model considered by Cho et al. (2015) and show that different from the results reported by these authors removing the cycle is always welfare improving. Welfare measures computed from weighted residuals methods confirm the logic behind our perturbation approach and verify the accuracy of our estimates.

The infinite extendibility problem for exchangeable real-valued random vectors

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on $\mathbb{R}^d$, which is: Can a given random vector $\vec{X} = (X_1,\ldots,X_d)$ be represented in distribution as the first $d$ members of an infinite exchangeable sequence of random variables? This is the case if and only if $\vec{X}$ has a stochastic representation that is "conditionally iid" according to the seminal de Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model $\vec{X}$ is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when $\vec{X}$ takes values in $\{0,1\}^d$ has a spherical law, a law with $\ell_1$-norm symmetric survival function, or a law with $\ell_{\infty}$-norm symmetric density. The solutions in all These cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on $\mathbb{R}$, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when $\vec{X}$ has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a L\'evy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems.

Expected number of nodal components for cut‐off fractional Gaussian fields

Let $({\mathcal{X}},g)$ be a closed Riemmanian manifold of dimension $n>0$. Let $\Delta$ be the Laplacian on ${\mathcal{X}}$, and let $(e\_k)\_k$ be an $L^2$-orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues $(\lambda\_k)\_k$. We assume that $(\lambda\_k)\_k$ is non-decreasing and that the $e\_k$ are real-valued. Let $(\xi\_k)\_k$ be a sequence of iid $\mathcal{N}(0,1)$ random variables. For each $L>0$ and $s\in{\mathbb{R}}$, possibly negative, set\[f^s\_L=\sum\_{0<\lambda\_j\leq L}\lambda\_j^{-\frac{s}{2}}\xi\_je\_j\, .\]Then, $f\_L^s$ is almost surely regular on its zero set. Let $N\_L$ be the number of connected components of its zero set. If $s<\frac{n}{2}$, then we deduce that there exists $\nu=\nu(n,s)>0$ such that $N\_L\sim \nu {Vol}\_g({\mathcal{X}})L^{n/2}$ in $L^1$ and almost surely. In particular, ${\mathbb{E}}[N\_L]\asymp L^{n/2}$. On the other hand, we prove that if $s=\frac{n}{2}$ then\[{\mathbb{E}}[N\_L]\asymp \frac{L^{n/2}}{\sqrt{\ln\left(L^{1/2}\right)}}\, .\]In the latter case, we also obtain an upper bound for the expected Euler characteristic of the zero set of $f\_L^s$ and for its Betti numbers. In the case $s>n/2$, the pointwise variance of $f\_L^s$ converges so it is not expected to have universal behavior as $L\rightarrow+\infty$.

Robust regression analysis for a censored response and functional regressors

ABSTRACTLet be an independent and identically distributed (iid) sequence of interest random variables (rv) distributed as T. In censorship models, T is subject to random censoring by another rv C. Based on the so-called synthetic data, we define an M-estimator for the regression function of T given a functional covariate . Under standard assumptions on the kernel, bandwidth and small ball probabilities, we establish its strong consistency with rate and asymptotic normality. The asymptotic variance is given explicitly. Confidence bands are given and special cases are studied to show the generality of our work. Finally simulations are drawn to illustrate both quality of fit and robustness.

Applications of distance correlation to time series

The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s (see for example, Feuerverger and Mureika (1977), Csorgo (1981a,1981b,1981c), Feuerverger (1993)). More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Szekely and Rizzo (2009) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an iid sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of different contexts.

Independence times for iid sequences, random walks and Lévy processes

For a sequence in discrete time having stationary independent values (respectively, random walk) X, those random times R of X are characterized set-theoretically, for which the strict post-R sequence (respectively, the process of the increments of X after R) is independent of the history up to R. For a Lévy process X and a random time R of X, reasonably useful sufficient conditions and a partial necessary condition on R are given, for the process of the increments of X after R to be independent of the history up to R.

On the Use of Second and Third Moments for the Comparison of Linear Gaussian and Simple Bilinear White Noise Processes

The linear Gaussian white noise process (LGWNP) is an independent and identically distributed (iid) sequence with zero mean and finite variance with distribution . Some processes, such as the simple bilinear white noise process (SBWNP), have the same covariance structure like the LGWNP. How can these two processes be distinguished and/or compared? If is a realization of the SBWNP. This paper studies in detail the covariance structure of . It is shown from this study that; 1) the covariance structure of is non-normal with distribution equivalent to the linear ARMA(2, 1) model; 2) the covariance structure of is iid; 3) the variance of can be used for comparison of SBWNP and LGWNP.

A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $\Xi $-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.

A note on the limit theory of a Dickey–Fuller unit root test with heavy tailed innovations

We are occupied with the limit theory of the OLSE and of a subsequent Dickey–Fuller test when the unit root process has heavy tailed and dependent innovations that do not possess moments of order α for some α∈0,2. The innovation process has the form of a “martingale-type” transform constructed as a pointwise product between an iid sequence in the domain of attraction of an α stable distribution with a non existing α moment, for some α∈0,2, and a positive scaling mixing sequence that has a slowly varying at infinity truncated α moment. We derive a functional limit theorem with complex rates and limits that depend on Levy α-stable processes. The OLSE remains superconsistent with rate n, and the limiting distribution is a functional of the previous process. When α=2 we recover the standard Dickey–Fuller distribution.

Some Applications of Higher Moments of the Linear Gaussian White Noise Process

The Linear Gaussian white noise process is an independent and identically distributed (iid) sequence with zero mean and finite variance with distribution N (0, σ2 ) . Hence, if X1, x2, …, Xn is a realization of such an iid sequence, this paper studies in detail the covariance structure of X1d, X2d, …, Xnd, d=1, 2, …. By this study, it is shown that: 1) all powers of a Linear Gaussian White Noise Process are iid but, not normally distributed and 2) the higher moments (variance and kurtosis) of Xtd, d=2, 3, … can be used to distinguish between the Linear Gaussian white noise process and other processes with similar covariance structure.

Martingale Transforms with Mixed Stable Limits and the QMLE for Conditionally Heteroskedastic Models

We derive a limit theorem for appropriately centered and scaled martingale transforms $\sum_{i=1}^{n}\xi_{i}V_{i}$ to mixed-stable limits when $\left(\xi_{i}\right)$ is an iid sequence in the domain of attraction of an $\alpha$-stable distribution where $\alpha\in(0,2]$. Using the Principle of Conditioning we recover and extend known results in the literature while imposing weaker conditions. The results are particularly useful in determining the limit theory of the Gaussian QMLE in conditionally heteroskedastic models when the squared innovations are heavy-tailed. We provide the framework for the QMLE limit theory which in the ergodic case is based on the stochastic recurrence approach used in the relevant literature and we furthermore allow for the parameter vector to lie on the boundary of the parameter space. Then we show that the QMLE weakly converges to an $\alpha$-stable distribution when $\alpha\in[1,2]$ and is inconsistent when $\alpha

On the reflected random walk on R+

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ = ∫Rydρ(y) &lt; 0. Consider the random walk on R+ starting at x ∈ R+ and defined for any n ∈N by \begin{eqnarray*} \left\{\begin{array}{rl} X_0&amp;=x \\ X_{n+1} &amp; = |X_n+Y_{n+1}| \end{array}\right. \end{eqnarray*} where (Yn) is an iid sequence of law ρ. We denote P the Markov operator associated to this random walk and, for any borelian bounded function f on R+, we call Poisson’s equation the equation f = g − Pg with unknown function g. In this paper, we prove that under a regularity condition on ρ and f, there is a solution to Poisson’s equation converging to 0 at infinity. Then, we use this result to prove the functional central limit theorem and it’s almost-sure version.

Nonparametric Prediction Intervals of generalized order statistics from two independent sequences

This paper, discusses the problem of predicting future a generalized order statistic of an iid sequence sample was drawn from an arbitrary unknown distribution, based on observed also generalized order statistics from the same population. The coverage probabilities of these prediction intervals are exact and free of the parent distribution F(). Prediction formulas of ordinary order statistics and upper record values are extracted as special cases from the productive results. Finally, numerical computations on several models of ordered random variables are given to illustrate the proposed procedures.

Martingale Transforms with Mixed Stable Limits and the QMLE for Conditionally Heteroskedastic Models

We derive a limit theorem for appropriately centered and scaled martingale transforms \sum_{i=1}^{n}\xi_{i}V_{i} to mixed-stable limits when \left(\xi_{i}\right) is an iid sequence in the domain of attraction of an \alpha-stable distribution where \alpha\in(0,2]. Using the Principle of Conditioning we recover and extend known results in the literature while imposing weaker conditions. The results are particularly useful in determining the limit theory of the Gaussian QMLE in conditionally heteroskedastic models when the squared innovations are heavy-tailed. We provide the framework for the QMLE limit theory which in the ergodic case is based on the stochastic recurrence approach used in the relevant literature and we furthermore allow for the parameter vector to lie on the boundary of the parameter space. Then we show that the QMLE weakly converges to an \alpha-stable distribution when \alpha\in[1,2] and is inconsistent when \alpha

Extremal dichotomy for uniformly hyperbolic systems

We consider the extreme value theory of a hyperbolic toral automorphism showing that, if a Hölder observation φ is a function of a Euclidean-type distance to a non-periodic point ζ and is strictly maximized at ζ, then the corresponding time series {φ○Ti} exhibits extreme value statistics corresponding to an independent identically distributed (iid) sequence of random variables with the same distribution function as φ and with extremal index one. If, however, φ is strictly maximized at a periodic point q, then the corresponding time-series exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as φ but with extremal index not equal to one. We give a formula for the extremal index, which depends upon the metric used and the period of q. These results imply that return times to small balls centred at non-periodic points follow a Poisson law, whereas the law is compound Poisson if the balls are centred at periodic points.

Asymptotic independence of three statistics of maximal segmental scores

Let ξ1,ξ2,… be an iid sequence with negative mean. The (m,n)-segment is the subsequence ξm+1,…,ξn and its score is given by max{∑m+1nξi,0}. Let Rn be the largest score of any segment ending at time n, Rn∗ the largest score of any segment in the sequence ξ1,…,ξn, and Ox the overshoot of the score over a level x at the first epoch the score of such a size arises. We show that, under the Cramér assumption on ξ1, asymptotic independence of the statistics Rn, Rn∗−y and Ox+y holds as min{n,y,x}→∞. Furthermore, we establish a novel Spitzer-type identity characterising the limit law O∞ in terms of the laws of (1,n)-scores. As corollary we obtain: (1) a novel factorisation of the exponential distribution as a convolution of O∞ and the stationary distribution of R; (2) if y=γ−1logn (where γ is the Cramér coefficient), our results, together with the classical theorem of Iglehart (1972), yield the existence and explicit form of the joint weak limit of (Rn,Rn∗−y,Ox+y).