This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of conditions for convergence may be reduced to one basic set, where conditions are given in terms of conditional moments of truncated variables, given the past. In the second part (sections 4 and 5) we first recall some basic facts from the modern "French" theory of stochastic processes, then show that Rebolledo's recent functional limit theorems for martingales with continuous time can be deduced from the limit theorems for discrete time martingales. Again, several related sets of conditions for convergence are formulated.

https://www.jstor.org/stable/info/4615860

Central Limit Theorems for Martingales with Discrete or Continuous Time

We consider a finite controlled Markov chain, the description of which depends on an unknown parameter a, and investigate the following control policy. To each a an optimal stationary control is associated. a is estimated recurrently from the trajectory by the minimum contrast method, and the optimal stationary control corresponding to the estimate is used. We present asymptotic properties of the estimate and of the criterion function. They follow from the law of large numbers and from the central limit theorem for controlled Markov chains derived with the aid of martingales.

Estimation and control in Markov chains

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for the U-statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.

Short distances, flat triangles and Poisson limits

The first result is a generalization of the classical results for independent random variables. The second result gives general conditions for convergence to processes which can be obtained from Brownian motion by a random change of time. This result is used to give a unified development of most of the martingale central limit theorems in the literature. An important aspect of our methods is that after the initial result is shown, we can avoid any further consideration of tightness.

/pdf/functional-limit-theorems-for-dependent-variables-43lujqunpj.pdf

Functional Limit Theorems for Dependent Variables

In the application of autoregressive models the order of the model is often estimated using either a sequence of likelihood ratio tests, a likelihood based information criterion, or a residual based test. The properties of such procedures has been discussed extensively under the assumption that the characteristic roots of the autoregression are stationary. While non-stationary situations have also been considered the results in the literature depend on conditions to the characteristic roots. It is here shown that these methods for lag length determination can be used regardless of the assumption to the characteristic roots and also in the presence of deterministic terms. The proofs are based on methods developed by C. Z. Wei in his joint work with T. L. Lai.

/pdf/order-determination-in-general-vector-autoregressions-3yqojm88jv.pdf

Order determination in general vector autoregressions

Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely divisible laws with bounded variances, of the row sums of elementary systems of independent random variables.

/pdf/martingale-convergence-to-infinitely-divisible-laws-with-4u1z28vl54.pdf

G. K. Eagleson

Papers

Martingale convergence to infinitely divisible laws with finite variances