Finite Variance Assumption Research Articles

Assessment of background noise properties in time and time–frequency domains in the context of vibration-based local damage detection in real environment

Any measurement in condition monitoring applications is associated with disturbing noise. Till now, most of the diagnostic procedures have assumed the Gaussian distribution for the noise. This paper shares a novel perspective to the problem of local damage detection. The acquired vector of observations is considered as an additive mixture of signal of interest (SOI) and noise with strongly non-Gaussian, heavy-tailed properties, that masks the SOI. The distribution properties of the background noise influence the selection of tools used for the signal analysis, particularly for local damage detection. Thus, it is extremely important to recognize and identify possible non-Gaussian behavior of the noise. The problem considered here is more general than the classical goodness-of-fit testing. The paper highlights the important role of variance, as most of the methods for signal analysis are based on the assumption of the finite-variance distribution of the underlying signal. The finite variance assumption is crucial but implicit to most indicators used in condition monitoring (such as the root-mean-square value, the power spectral density, the kurtosis, the spectral correlation, etc.), in view that infinite variance implies moments higher than 2 are also infinite. The problem is demonstrated based on three popular types of non-Gaussian distributions observed for real vibration signals. We demonstrate how the properties of noise distribution in the time domain may change by its transformations to the time–frequency domain (spectrogram). Additionally, we propose a procedure to check the presence of the infinite-variance of the background noise. Our investigations are illustrated using simulation studies and real vibration signals from various machines.

Correlation versus co-fractality: Evidence from foreign-exchange-rate variances

The concept of correlation appears to be the cornerstone of modern finance as it is applied in almost all finance-related research studies. However, Fama (1963) argued that “if the [population] variance is infinite, other statistical tools (e.g., least-squares regression) which are based on the assumption of finite variance will, at best, be considerably weakened and may in fact give very misleading answers” (p. 421). This study shows variances of foreign exchange rates to be governed by power laws with a tail exponent of α < 3, suggesting infinite second moments. We derive a new concept to measure dependencies between power-law processes with this tail exponent, which we term co-fractality. We show that risk diversification based on the concept of correlation indeed gives misleading results. Notably, foreign-exchange-rate variances lacking co-fractality in our earlier subsample do not show evidence for co-fractality in our later subsample. We argue that co-fractality, as opposed to correlation, should be used to measure the dependency between processes governed by power laws.

Asymptotic normality and confidence region for Catoni's Z estimator

This paper studies the asymptotic property and confidence region of Catoni's Z estimator under the finite variance assumption. First, we investigate the CLT for Catoni's Z estimator with the asymptotically optimal variance, if the tuning parameter has order . Second, we propose the normal approximated confidence region of Catoni's Z estimator. The simulations show that the proposed 99% CLT‐based confidence regions are shorter than the existing non‐asymptotic confidence regions; mean squared errors of Catoni's Z estimators are typically smaller than sample means from heavy‐tailed data. Furthermore, the simulated coverage probabilities of the CLT‐based confidence region are close to the confidence levels when the sample size is moderately large. For 6‐year DOCVIS data (the number of doctor visits) in the German health care demand dataset, our CLT‐based confidence regions are tighter than non‐asymptotic confidence regions.

A generalized Catoni’s M-estimator under finite α-th moment assumption with α∈(1,2)

We generalize Catoni’s M-estimator, put forward in [3] by Catoni under finite variance assumption, to the case in which distributions can have finite α-th moment with α∈(1,2). Our approach, inspired by the Taylor-like expansion developed in [4], is via slightly modifying the influence function φ in [3]. A deviation bound is established for this generalized estimator, and coincides with that in [3] as α↑2. Experiment shows that our M-estimator performs better than the empirical mean, the smaller the α is, the better the performance will be. As an application, we study an ℓ1 regression considered by Zhang et al. [19], who assumed that samples have finite variance, under finite α-th moment assumption with α∈(1,2).

Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schrödinger equation (see work of Zakharov, Glassey, and Ogawa–Tsutsumi) to show that certain classes of negative energy solutions must blowup in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.

A new direct proof of the central limit theorem

We prove the central limit theorem from the definition of weak convergence using the Haar basis, calculus, and elementary probability, and we estimate the rate of convergence off the tails. The use of the Haar basis pinpoints the role of $L^{2}([0,1])$ in the CLT as well as the assumption of finite variance.

Stable limit theorem for $U$-statistic processes indexed by a random walk

Let $(S_n)_{n\in \mathbb{N} }$ be a $\mathbb{Z} $-valued random walk with increments from the domain of attraction of some $\alpha $-stable law and let $(\xi (i))_{i\in \mathbb{Z} }$ be a sequence of iid random variables. We want to investigate $U$-statistics indexed by the random walk $S_n$, that is $U_n:=\sum _{1\leq i<j\leq n}h(\xi (S_i),\xi (S_j))$ for some symmetric bivariate function $h$. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the $U$-statistic $U_n$.

Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

Portmanteau tests for ARMA models with infinite variance

Abstract. Autoregressive and moving‐average (ARMA) models with stable Paretian errors are some of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking of fitted models has not been addressed. In this article, we develop portmanteau tests for checking the randomness of a time series with infinite variance and for ARMA diagnostic checking when the innovations have infinite variance. It is assumed that least squares or an asymptotically equivalent estimation method, such as Gaussian maximum likelihood, is used. It is also assumed that the distribution of the innovations is identically and independently distributed (i.i.d.) stable Paretian. It is seen via simulation that the proposed portmanteau tests do not converge well to the corresponding limiting distributions for practical series length so a Monte Carlo test is suggested. Simulation experiments show that the proposed Monte Carlo test procedure works effectively. Two illustrative applications to actual data are provided to demonstrate that an incorrect conclusion may result if the usual portmanteau test based on the finite variance assumption is used.

Tanaka Formulae for ( , d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (α, d, β)-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L1+θ convergence argument (with 0<θ<β≤1) rather than L2 convergence, for the derivation of the Tanaka formulae.

Testing for Mean Reversion in Processes of Ornstein-Uhlenbeck Type

In this paper, we analyse processes of Ornstein-Uhlenbeck (OU) type, driven by Levy processes. This class is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Levy process with no OU (autoregressive) effect. For an appropriate discretised version of the model, we utilise likelihood methods to test for such a reduction of the OU process to Levy motion, deriving the distribution of the relevant pseudo-log-likelihood ratio statistics, asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as the length of the observation window grows large. These analyses are non-standard in that the mean reversion parameter vanishes under the null of a pure Levy process for the data. Despite this we are able to give a very general analysis with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Levy process. As a special case, for Brownian motion as driving process, we deduce the limiting distribution in a quite explicit way, finding results which generalise the well-known Dickey-Fuller (‘unit-root’) theory.

Asymptotic Properties of Least Squares Estimators for the Parameters in Undamped Exponential Signals

SYNOPTIC ABSTRACTThe superimposed exponential signals model plays an important role in Statistical Signal Processing. In this paper we study the properties of the least squares estimators and the estimators obtained by maximizing the periodogram. The asymptotic equivalence of these two estimators can be obtained. The consistency and the asymptotic distribution of the least squares estimators under the assumption of finite variance of the error random variables are established. Some simulation results are reported comparing the performances of the two different estimators.

The consistency of the L1norm estimates in arma models

The L1-norm estimates of the ARMA model parameters are shown to be strongly consistent via a recent result obtained by Dupacova and Wets and based on nonsmooth analysis. Moreover, a stationarity and invertibility condition avoiding the usual assumption of finite variance is given and some properties of the stationarity and invertibility region are presented.

On the null recurrence and transience of a first-order SETAR model

We consider the SETAR(l; 1, ···, 1) model: where – ∞= r 0 &amp;lt; r 1, &amp;lt; · ·· &amp;lt; rl = ∞ and for each j {ε t (j)} forms an i.i.d. zero-mean error sequence independent of {ε t (i)} for i ≠ j and having a density positive on the real line. Chan et al. (1985) obtained the region of the parameter space on which the process is ergodic, and showed the process to be transient on a subset of the remainder. They conjectured that the process was null recurrent everywhere else. In this paper we show that conjecture to be incorrect and under the assumption of finite variance of the error distributions we resolve the remaining questions of transience or null recurrence for this process.

On the null recurrence and transience of a first-order SETAR model

We consider the SETAR(l; 1, ···, 1) model: where – ∞= r0 &lt; r1, &lt; · ·· &lt; rl = ∞ and for each j {ε t(j)} forms an i.i.d. zero-mean error sequence independent of {ε t(i)} for i ≠ j and having a density positive on the real line. Chan et al. (1985) obtained the region of the parameter space on which the process is ergodic, and showed the process to be transient on a subset of the remainder. They conjectured that the process was null recurrent everywhere else. In this paper we show that conjecture to be incorrect and under the assumption of finite variance of the error distributions we resolve the remaining questions of transience or null recurrence for this process.

A Renewal Theorem for Random Walks in Multidimensional Time

Let $X, {X_1}, {X_2}, \ldots$ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + {X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers ${n_j}, 1 \leq j \leq r, R(k)$ becomes the renewal function $Q(k)$ for a random walk in $r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum \nolimits _{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n), 1 \leq n \leq x$, which by itself is of interest.

A renewal theorem for random walks in multidimensional time

Let $X, {X_1}, {X_2}, \ldots$ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + {X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers ${n_j}, 1 \leq j \leq r, R(k)$ becomes the renewal function $Q(k)$ for a random walk in $r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum \nolimits _{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n), 1 \leq n \leq x$, which by itself is of interest.

On normal characterizations by the distribution of linear forms, assuming finite variance

If X 1 and X 2 are independent and identically distributed (i. i. d.) with finite variance, then (X 1+X 2) √2 has the same distribution as X 1 if and only if X 1 is normal with mean zero (Pólya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where σ ∞ i=1 a i X i has the same distribution as X 1. In particular if at least two of the a i's are non-zero and X 1 has finite variance, then Laha and Lukacs [8] showed that X 1 is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Pólya in [9]. The same technique is also used to give a characterization of the exponential distribution.