Conditional Central Limit Theorem Research Articles

Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes

This paper considers methods of deriving sufficient conditions for the central limit theorem and functional central limit theorem to hold in a broad class of time series processes, including nonlinear processes and semiparametric linear processes. The common thread linking these results is the concept of near-epoch dependence on a mixing process, since powerful limit results are available under this limited-dependence property. The particular case of near-epoch dependence on an independent process provides a convenient framework for dealing with a range of nonlinear cases, including the bilinear, GARCH, and threshold autoregressive models. It is shown in particular that even SETAR processes with a unit root regime have short memory, under the right conditions. A simulation approach is also demonstrated, applicable to cases that are analytically intractable. A new FCLT is given for semiparametric linear processes, where the forcing processes are of the NED-on-mixing type, under conditions that are evidently close to necessary.

Conditional Studentized Survival Tests for Randomly Censored Models

It is shown that in the case of heterogenous censoring distributions Studentized survival tests can be carried out as conditional permutation tests given the order statistics and their censoring status. The result is based on a conditional central limit theorem for permutation statistics. It holds for linear test statistics as well as for sup‐statistics. The procedure works under one of the following general circumstances for the two‐sample problem: the unbalanced sample size case, highly censored data, certain non‐convergent weight functions or under alternatives. For instance, the two‐sample log rank test can be carried out asymptotically as a conditional test if the relative amount of uncensored observations vanishes asymptotically as long as the number of uncensored observations becomes infinite. Similar results hold whenever the sample sizes and are unbalanced in the sense that and hold.

Central limit theorems for quadratic forms with time-domain conditions

We establish the central limit theorem for quadratic forms $\Sigma_{t, s=1}^N b(t - s)P_{m, n} (X_t, X_s)$ of the bivariate Appell polynomials $P_{m, n} (X_t, X_x))$ under time-domain conditions. These conditions relate the weights $b(t)$ and the covariances of the sequences $(P_{m, n} (X_t, X_s))$ and $(X_t)$. The time-domain approach, together with the spectral domain approach developed earlier, yields a general set of conditions for central limit theorems.

Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem

It is shown that permutation tests based on studentized statistics are asymptotically exact of size α also under certain extended non-i.i.d. null hypotheses. To demonstrate the principle the results are applied to the generalized two-sample Behrens-Fisher problem for testing equality of the means under general non-parametric heterogeneous error distributions. Within this setting we propose a permutation version of the Welch test which is an extension of Pitman's two-sample permutation test. These results are special cases of a conditional central limit theorem for studentized permutation statistics which also applies to asymptotic power functions.

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure $P$ and a random (a.s.) linear functional $F$ on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of $P$ under $F$ are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

Large Deviations for Large Capacity Loss Networks withFixed Routing and Polyhedral Admission Sets

In this paper, we study large deviations of large capacity loss networks with fixed routing. We use two-level modelling for the loss networks: the call level and the cell level. At the call level, a call request is accepted if it succeeds an admission test. The test is based on a polyhedral set of the number of calls in progress when a new call arrives. After being accepted, a call then transmits a sequence of cells (random variables) during its holding period. We show that the fluid limits and the conditional central limit theorems in Kelly (1991) can be extended to the large deviation regime. Moreover, there are corresponding fluid flow explanations for our large deviation results. In particular, we derive the exponential decay rates of the call blocking probability and the cell loss probability. These decay rates are obtained by solving primal and dual convex programming problems.

An almost sure central limit theorem for the overlap parameters in the Hopfield model

We consider the Hopfield model with a finite number of randomly chosen patterns above and below the critical temperature and prove an almost sure conditional central limit theorem for the vector of overlap parameters. For this purpose we analyse the almost sure asymptotic behaviour of the partition function.

Asymptotic expansions in the conditional central limit theorem

Let X n , n∈ N , be i.i.d. with mean 0, variance 1, and E(¦X n¦ r) < ∞ for some r ⩾ 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P( 1 √n Σ i = 1 n X i ⩽ t¦B ) can be approximated by a modified Edgeworth expansion up to order o( 1 n (r − 2) 2) ) , if the distances of the set B from the σ-fields σ( X 1, …, X n ) are of order O( 1 n (r − 2) 2) ( lg n) β), where β < −(r − 2) 2 for r∉ N and β < −r 2 for r∈ N . An example shows that if we replace β < −(r − 2) 2 by β = −(r − 2) 2 for r∉ N (β < −r 2 by β = −r 2 for r∈ N ) we can only obtain the approximation order O( 1 n (r − 2) 2) ) for r∉ N (O( lg lg n n (r − 2) 2) ) for r∈ N ).

Nonuniform Estimates in the Conditional Central Limit Theorem

Let $X_n, n \in \mathbb{N}$, be i.i.d. with mean 0, variance 1, and $E(|X_1|^r) 3$. Let $B$ be a measurable set such that its distances from the $\sigma$ fields $\sigma (X_1,\ldots, X_n)$ are of order $O(n^{-1/2} (\log n)^{-r/2})$. We prove that for such $B$ the conditional probabilities $P(n^{-1/2}\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by the standard normal distribution $\Phi (t)$ up to the classical nonuniform bound $(1 + |t|^r)^{-1} n^{-1/2}$. An example shows that this is not true any more if the distances of $B$ from $\sigma (X_1,\ldots, X_n)$ are only of order $O(n^{-1/2}(\log n)^{-r/2+\varepsilon})$ for some $\varepsilon > 0$. For the case $r = 3$ one can obtain the corresponding assertion only under a strengthened assumption.

An asymptotically efficient difference formula for solving stochastic differential equations

Time discretisations of teh vector stochastic differential equation are considered, where (y t) is a continuous scalar process whose distribution is absolutely continuous with respect to Wiener measure. Among approximations to x T that depend on (y t) only at the discretisation points, the conditional mean is asymptotically optimal in the sense that it minimises all symmetrical conditional moments of the error. A conditional central limit theorem is derived for this minimal error, and a finite difference formula is developed which yields appromimations with the same asymptotically optimal properties. This formula is necessarily more complex than the familiar Milshtein scheme (Milshtein [7]). The latter has the maximum order of convergence but its error, considered as a power serioes in the discretisation parameter h, does not have the minimal leading coefficient. The results generalise for a special class of equations with multi-dimensional forcing terms

Second-Order Approximation in the Conditional Central Limit Theorem

Let $X_n, n \in \mathbb{N}$ be i.i.d. with mean 0 and variance 1. Let $B \in \sigma(X_n: n \in \mathbb{N})$ be a set such that its distances from the $\sigma$-fields $\sigma(X_1,\cdots, X_n)$ are of order $O(1/n(lg n)^{2 + \varepsilon})$ for some $\varepsilon > 0$. We prove that for those $B$ the conditional probabilities $P((1/\sqrt n)\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by a modified Edgeworth expansion up to order $O(1/n)$. An example shows that this is not true any more if the distances of $B$ from $\sigma(X_1,\cdots, X_n)$ are only of order $O(1/n(lg n)^2)$.

Remainder term estimates in a conditional central limit theorem for integer-valued random variables

AbstractA Berry-Esseen type result is given for the conditional distribution of a weighted sum of i.i.d. integer-valued r.v.'s given that their unweighted sum equals its expectation. The examples include the case of sampling without replacement from a finite population.

Exact approximation orders in the conditional central limit theorem

Exact approximation orders in the conditional central limit theorem

Correction Note: Correction to "New Conditions for Central Limit Theorems"

Correction Note: Correction to "New Conditions for Central Limit Theorems"

New Conditions for Central Limit Theorems

Abstract : A general formulation of the central limit problem for sums of independent random variables is given. By assuming the existence of fourth- order moments, one is able to prove new necessary and sufficient conditions for both Normal and Poisson convergence which involve only moments. The proof of the theorem makes use of a characterization of the Normal distribution among infinitely divisible laws which was perhaps first recognized by Borges and later independently by the author.