Stationary Martingale Differences Research Articles

The central limit theorem for random perturbations of rotations

We prove a functional central limit theorem for stationary random sequences given by the transformations on the two-dimensional torus. This result is based on a functional central limit theorem for ergodic stationary martingale differences with values in a separable Hilbert space of square integrable functions.

A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

We consider a stationary time series {Xt} given byXt=∑∞k=−∞ψkZt−k, where {Zt} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {Xt} is squared integrable andmτ∑|k|⩾mψ2k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionmτ∑|k|⩾mψ2k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm∑|k|⩾mψ2k→0.

A law of the iterated logarithm for geometrically weighted martingale difference sequences

We consider random variables ξ(β)=Σn=0∞βnYn for β<1. We prove that if the(Yn)n∈N is a stationary ergodic martingale difference sequence andEY02=1, then the following law of the iterated logarithm holds: $$\mathop {\lim \sup }\limits_{\beta \nearrow 1} \frac{{\sqrt {1 - \beta ^2 } }}{{\sqrt {2\log \log \frac{1}{{1 - \beta ^2 }}} }}\xi (\beta ) = 1 a.s.$$ We prove also the corresponding Central Limit Theorem. This generalizes a theorem by Bovier and Picco where the i.i.d. case was studied.

An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

In this paper we establish an almost sure invariance principle with an error termo((t log logt)1/2) (ast→∞) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as o-mixing and α-mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.

Asymptotic properties of time domain gaussian estimators

We consider a general non-linear multivariate time series model which can be parameterized by a finite and fixed number of parameters and which can be rewritten, if necessary, in a form such that the disturbances are stationary martingale differences. Given a series of discrete, equally spaced observations we prove the strong consistency and asymptotic normality of the Gaussian estimators of the parameters, the parameters possibly being subject to non-linear constraints. Because the normal equations are usually highly non-linear it may be difficult to obtain explicit expressions for the Gaussian estimates. To overcome this problem we use a Gauss–Newton type algorithm to obtain a sequence of iterates which converge to, and have the same asymptotic properties as, the Gaussian estimates.

Asymptotic properties of time domain gaussian estimators

We consider a general non-linear multivariate time series model which can be parameterized by a finite and fixed number of parameters and which can be rewritten, if necessary, in a form such that the disturbances are stationary martingale differences. Given a series of discrete, equally spaced observations we prove the strong consistency and asymptotic normality of the Gaussian estimators of the parameters, the parameters possibly being subject to non-linear constraints. Because the normal equations are usually highly non-linear it may be difficult to obtain explicit expressions for the Gaussian estimates. To overcome this problem we use a Gauss–Newton type algorithm to obtain a sequence of iterates which converge to, and have the same asymptotic properties as, the Gaussian estimates.

The Convergence of Some Recursions

In connection with a range of stationary time series models, particularly ARMAX models, recursive calculations of the parameter vector seem important. In these the estimate, $\theta(n)$, from observations to time $n$, is calculated as $\theta(n) = \theta(n - 1) + k_n$ where $k_n$ depends only on $\theta(n - 1), \theta(n - 2), \cdots$ and the data to time $n$. The convergence of two recursions is proved for the simple model $x(n) = \varepsilon(n) + \alpha\varepsilon(n - 1), |\alpha| < 1$, where the $\varepsilon(n)$ are stationary ergodic martingale differences with $E\{\varepsilon(n)^2\mid\mathscr{F}_{n-1}\} = \sigma^2$. The method of proof consists in reducing the study of the recursion to that of a recursion involving the data only through the $\theta(n)$. It seems that many of the recursions introduced for ARMAX models may be treated in this way and the nature of the extensions of the theory is discussed.

On the central limit theorem and iterated logarithm law for stationary processes

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.