Sum Of Processes Research Articles

Convergence of functionals of sums of r.v.s to local times of fractional stable motions

Consider a sequence Xk=∑j=0∞cjξk−j, k≥1, where cj, j≥0, is a sequence of constants and ξj, −∞<j<∞, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<α≤2. Let Sk=∑j=1kXj. Under suitable conditions on the constants cj it is known that for a suitable normalizing constant γn, the partial sum process γn−1S[nt] converges in distribution to a linear fractional stable motion (indexed by α and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process Xk. In this paper it is established that the process n−1βn∑k=1[nt]f(βn(γn−1Sk+x)) converges in distribution to ( ∫−∞∞f( y) dy)L(t,−x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f( y) that includes the indicator functions of bounded intervals of the real line. Here βn→∞ such that n−1βn→0. The only further condition that is assumed on the distribution of ξ1 is that either it satisfies the Cramér’s condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.

Some Useful Functions for Functional Large Deviations

Useful descriptions of stochastic models are often provided when they are represented as functions of well understood stochastic models. Properties of the well understood model can be preserved by the representation. For example, the large deviation principle (LDP) is preserved by continuous maps via the contraction principle and weak convergence is preserved by continuous maps via the continuous mapping theorem. In a recent paper, Ganesh and O'Connell demonstrate the value of particular topological space of functions indexed by the positive real line by showing: (1) a large class of partial sums process satisfy the functional LDP in the space; and (2) the supremum map is continuous when restricted to an appropriate subspace. This makes the space useful, for example, in proving logarithmic asymptotics for the stationary waiting-time distribution at a single server queue. This paper facilitates further deductions from the space by considering other useful maps: inversion, composition and first passage time. It is shown that composition and inversion are continuous when restricted to an appropriate subspace. First passage time is not continuous, but with additional rate-function assumptions the LDP can be deduced by explicit calculation. A number of examples of these results are presented.

REGRESSION MODEL FITTING WITH A LONG MEMORY COVARIATE PROCESS

This paper proposes some tests for fitting a regression model with a long memory covariate process and with errors that form either a martingale difference sequence or a long memory moving average process, independent of the covariate. The tests are based on a partial sum process of the residuals from the fitted regression. The asymptotic null distribution of this process is discussed in some detail under each set of these assumptions. The proposed tests are shown to have known asymptotic null distributions in the case of martingale difference errors and also in the case of fitting a polynomial of a known degree through the origin when the errors have long memory. The theory is then illustrated with some examples based on the forward premium anomaly where a squared interest rate differential proxies a time dependent risk premium. The paper also shows that the proposed test statistic converges weakly to nonstandard distributions in some cases.The authors gratefully acknowledge the helpful comments of the co-editor Don Andrews and two anonymous referees. The research of the first two authors was partly supported by NSF grant DMS 00-71619.

Tabulating weighted sup-norm functionals used in change-point analysis

We introduce a number of weighted partial sum processes of which certain sup-norm functionals may be used, for example, to detect changes in the mean of independent observations. Their limiting distributions are derived mathematically, simulated and then tabulated. With this information, a detailed numerical investigation of the power of these functionals is carried out.*† *Research mainly done while at Carleton University, partially supported by an NSERC Canada Grant of Miklós Csörgő at Carleton University, Ottawa. Also supported by the Austrian Science Foundation (FWF) under grant SFB#010 (‘Adaptive Information Systems and Modelling in Economics and Management Science’) while at Technische Universität Wien. †Research partially supported by an NSERC Canada Grant of Miklós Csörgő at Carleton University, Ottawa.

Stable limits of sums of bounded functions of long-memory moving averages with finite variance

We discuss limit distributions of partial sums of bounded functions h of a long-memory moving-average process Xt= ∑j=1 ∞ bj ζt-j with coefficients bj decaying as j-β, 1/2< β< 1, and independent and identically distributed innovations ζs whose probability tails decay as x-α, 2< α< 4. The case of h having Appell rank k*=2 or 3 is discussed in detail. We show that in this case and in the parameter region αβ< 2 , the partial sums process, normalized by N1/αβ , weakly converges to an αβ-stable Lévy process, provided that the normalization dominates the corresponding k* th-order Hermite process normalization, or that 1/αβ> 1 - (2β-1)k*/2. A complete characterization of limit distributions of the partial sums process remains open.

Asymptotic behaviors for partial sum processes of a Gaussian sequence

We establish some large increment results for partial sum processes of a dependent stationary Gaussian sequence via estimating upper bounds of large deviation probabilities on suprema of the Gaussian sequence.

Random coefficient autoregression, regime switching and long memory

We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Lévy process.

Random coefficient autoregression, regime switching and long memory

We discuss long-memory properties and the partial sums process of the AR(1) process {Xt, t ∈ 𝕫} with random coefficient {at, t ∈ 𝕫} taking independent values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic Aj has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {Xt} decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {Xt} weakly converges to a stable Lévy process.

Module d'oscillation fonctionnel de quelques processus réels

In this Note we give the almost sure rate of uniform approach of Strassen functions by increments of empirical, quantile, Poisson, Wiener and Partial sums processes. Minimal conditions on the location of the increments are introduced and most useful critical limit functions are taken into account. To cite this article: P. Berthet, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Donsker's theorem for self-normalized partial sums processes

Let $X, X_1, X_2,\ldots$ be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if $\max_{1\le j\le n}|X_j|/V_n\to_P 0$, as $n\to\infty$, where $V_n^2=\sum_{j=1}^{n}X_j^2$.

Monitoring the evolutionary process of quality: risk-adjusted charting to track outcomes in intensive care.

To present graphical procedures for prospectively monitoring outcomes in the intensive care unit. Observational study: risk-adjusted control chart analysis of a case series. Tertiary referral adult intensive care unit: Princess Alexandra Hospital, Brisbane, Australia. A total of 3398 intensive care unit admissions from January 1, 1995, to January 1, 1998. Risk-adjusted process control charting procedures for continuous monitoring of intensive care unit outcomes are proposed as quality management tools. A modified Shewhart p chart and cumulative sum process control chart, using the Acute Physiology and Chronic Health Evaluation III model mortality prediction for risk adjustment, are presented. The risk-adjusted p chart summarizes performance at arbitrary intervals and plots observed against predicted mortality rate to detect large changes in risk-adjusted mortality. The risk-adjusted cumulative sum procedure is a likelihood-based scoring method that adjusts for estimated risk of death, accumulating evidence from outcomes of all previous patients. It formally tests the hypothesis of a change in the odds of death. In this application, we detected a decrease from above to predicted risk-adjusted mortality. This was temporally related to increased senior staffing levels and enhanced ongoing multidisciplinary review of practice, quality improvement, and educational activities. Formulas and analyses are provided as appendices.

A bootstrap approximation to a unit root test statistic for heavy-tailed observations

In the context of the AR(1) model with innovations in the domain of attraction of an α-stable law, we develop a residual bootstrap approximation to the distribution of a least-squares estimator of the autoregressive parameter when this parameter is equal to unity. Our procedure requires drawing bootstrap samples of size m< n, n being the size of the original sample. We establish the convergence in probability of the bootstrap distribution function assuming that m→∞ and m/ n→0. An analogous result is established for the partial sum process of the bootstrap noise sequence.

Limit Theorems for Maxima of Sums and Renewal Processes

Let {Xi}i=1,2,... be a sequence of i.i.d. random variables, let Sn = X1 + ... + Xn, and let Sn ← α a.s. We discuss necessary and sufficient conditions for the Kolmogorov and Marcinkiewicz–Zygmund type strong laws of large numbers and for the law of the iterated logarithm for renewal processes defined in two different ways. Bibliography: 16 titles.

A functional Hungarian construction for sums of independent random variables

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions f from a class H , but the supremum over f∈ H is taken outside the probability. This form is a prerequisite for the Komlós–Major–Tusnády inequality in the space of bounded functionals l ∞( H) , but contrary to the latter it essentially preserves the classical n −1/2log n approximation rate over large functional classes H such as the Hölder ball of smoothness 1/2. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.

External bootstrap tests for parameter stability

Abstract This article considers tests for parameter stability over time in general econometric models, possibly nonlinear-in-variables. Existing test statistics are commonly not asymptotically pivotal under nonstandard conditions. In such cases, the external bootstrap tests proposed in this paper are appealing from a practical viewpoint. We propose to use bootstrap versions of the asymptotic critical values based on a first-order asymptotic expansion of the test statistics under the null hypothesis, which consists of a linear transformation of the unobserved “innovations” partial sum process. The nature of these transformations under nonstandard conditions is discussed for the main testing principles. Also, we investigate the small sample performance of the proposed bootstrap tests by means of a small Monte Carlo experiment.

Limit Theorems for the Non-linear Functional of Stationary Gaussian Processes

In this paper we consider two functional limit theorems for the non-linear functional of the stationary Gaussian process satisfying short range dependence conditions: the functional CLT for partial sum processes and the uniform CLT for a special class of functions. To carry out the proofs, we develop Rosenthal type inequalities for the functional of Gaussian processes.

THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| &lt; ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| &lt; ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

Regular variation and the functional central limit theorem for heavy tailed random vectors

Multivariable regular variation is used, along with the martingale central limit theorem, to give a very simple proof that the partial sum process for a sequence of independent, identically distributed random vectors converges to a Brownian motion whenever the summands belong to the generalized domain of attraction of a normal law. This includes the heavy tailed case, where the covariance matrix is undefined because some of the marginals have infinite variance.

Infinitely Divisible Limit Processes for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature.