14 research outputs found
A free central-limit theorem for dynamical systems
The free central-limit theorem, a fundamental theorem in free probability,
states that empirical averages of freely independent random variables are
asymptotically semi-circular. We extend this theorem to general dynamical
systems of operators that we define using a free random variable coupled
with a group of *-automorphims describing the evolution of . We introduce
free mixing coefficients that measure how far a dynamical system is from being
freely independent. Under conditions on those coefficients, we prove that the
free central-limit theorem also holds for these processes and provide
Berry-Essen bounds. We generalize this to triangular arrays and U-statistics.
Finally we draw connections with classical probability and random matrix theory
with a series of examples
Recommended from our members
Limit theorems beyond sums of I.I.D observations
We consider second and third order limit theorems--namely central-limit theorems, Berry-Esseen bounds and concentration inequalities-- and extend them for "symmetric" random objects, and general estimators of exchangeable structures.
At first, we consider random processes whose distribution satisfies a symmetry property. Examples include exchangeability, stationarity, and various others. We show that, under a suitable mixing condition, estimates computed as ergodic averages of such processes satisfy a central limit theorem, a Berry-Esseen bound, and a concentration inequality. These are generalized further to triangular arrays, to a class of generalized U-statistics, and to a form of random censoring. As applications, we obtain new results on exchangeability, and on estimation in random fields and certain network model; extend results on graphon models; give a simpler proof of a recent central limit theorem for marked point processes; and establish asymptotic normality of the empirical entropy of a large class of processes. In certain special cases, we recover well-known properties, which can hence be interpreted as a direct consequence of symmetry. The proofs adapt Stein's method.
Subsequently, we consider a sequence of-potentially random-functions evaluated along a sequence of exchangeable structures. We show that, under general stability conditions, those values are asymptotically normal. Those conditions are vaguely reminiscent of those familiar from concentration results, however not identical. We require that the output of the functions does not vary significantly when an entry is disturbed; and the size of this variation should not depend markedly on the other entries.
Our result generalizes a number of known results, and as corollaries, we obtain new results for several applications: For randomly sub-sampled subgraphs; for risk estimates obtained by K-fold cross validation; and for the empirical risk of double bagging algorithms. The proof adapts the martingale central-limit theorem
Limit theorems for distributions invariant under groups of transformations
A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory implies a law of large numbers for such invariant distributions: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled averages, and to a generalization of U-statistics. As applications, we obtain a general limit theorem for exchangeable random structures, and new results on stationary random fields, network models, and a class of marked point processes. We also establish asymptotic normality of the empirical entropy for a large class of processes. Some known results are recovered as special cases, and can hence be interpreted as an outcome of symmetry. The proofs adapt Stein's method
Wasserstein-p Bounds in the Central Limit Theorem under Weak Dependence
The central limit theorem is one of the most fundamental results in
probability and has been successfully extended to locally dependent data and
strongly-mixing random fields. In this paper, we establish its rate of
convergence for transport distances, namely for arbitrary we obtain an
upper bound for the Wassertein- distance for locally dependent random
variables and strongly mixing stationary random fields. Our proofs adapt the
Stein dependency neighborhood method to the Wassertein- distance and as a
by-product we establish high-order local expansions of the Stein equation for
dependent random variables. Finally, we demonstrate how our results can be used
to obtain tail bounds that are asymptotically tight, and decrease polynomially
fast, for the empirical average of weakly dependent random variables
Wasserstein-p Bounds in the Central Limit Theorem Under Local Dependence
The central limit theorem (CLT) is one of the most fundamental results in
probability; and establishing its rate of convergence has been a key question
since the 1940s. For independent random variables, a series of recent works
established optimal error bounds under the Wasserstein-p distance (with p>=1).
In this paper, we extend those results to locally dependent random variables,
which include m-dependent random fields and U-statistics. Under conditions on
the moments and the dependency neighborhoods, we derive optimal rates in the
CLT for the Wasserstein-p distance. Our proofs rely on approximating the
empirical average of dependent observations by the empirical average of i.i.d.
random variables. To do so, we expand the Stein equation to arbitrary orders by
adapting the Stein's dependency neighborhood method. Finally we illustrate the
applicability of our results by obtaining efficient tail bounds.Comment: 49 pages. arXiv admin note: substantial text overlap with
arXiv:2209.0937
Asymptotics of Network Embeddings Learned via Subsampling
Network data are ubiquitous in modern machine learning, with tasks of
interest including node classification, node clustering and link prediction. A
frequent approach begins by learning an Euclidean embedding of the network, to
which algorithms developed for vector-valued data are applied. For large
networks, embeddings are learned using stochastic gradient methods where the
sub-sampling scheme can be freely chosen. Despite the strong empirical
performance of such methods, they are not well understood theoretically. Our
work encapsulates representation methods using a subsampling approach, such as
node2vec, into a single unifying framework. We prove, under the assumption that
the graph is exchangeable, that the distribution of the learned embedding
vectors asymptotically decouples. Moreover, we characterize the asymptotic
distribution and provided rates of convergence, in terms of the latent
parameters, which includes the choice of loss function and the embedding
dimension. This provides a theoretical foundation to understand what the
embedding vectors represent and how well these methods perform on downstream
tasks. Notably, we observe that typically used loss functions may lead to
shortcomings, such as a lack of Fisher consistency.Comment: 98 pages, 3 figures, 1 tabl
Smooth Edgeworth Expansion and Wasserstein- Bounds for Mixing Random Fields
The Edgeworth expansion is a central tool of probability that offers some
refinement on the central limit theorem by providing higher-order
approximations. In this paper, we consider -dimensional mixing random fields
and study the empirical average
. Firstly, under mixing and
moment conditions, we obtain a smooth Edgeworth expansion for to any
order . The proof relies on the Stein's method and a new constructive graph
approach. Then we utilize the obtained expansion and a newly proposed method in
Liu and Austern [2023] to obtain the first rates for the central limit theorem
in Wasserstein- distance for arbitrary . Finally, we apply those
results to obtain tail bounds and non-uniform Berry-Esseen bounds with
polynomial decay.Comment: 92 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:2209.0937