On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A central limit theorem for strongly multiplicative systems of functions

A necessary and sufficient condition is obtained for the validity of a central limit theorem for strongly multiplicative systems of functions.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 443–450, October, 1969.     

A non-commutative central limit theorem

This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123)     

A central limit theorem for integer partitions

Recently, Hwang proved a central limit theorem for restricted Λ-partitions, where Λ can be any nondecreasing sequence of integers tending to infinity that satisfies certain technical conditions. In particular, one of these conditions is that the associated Dirichlet series has only a single pole on the abscissa of convergence. In the present paper, we show that this condition can be relaxed, and provide some natural examples that arise from the study of integers with restrictions on their digital (base-b) expansion.     

A central limit theorem for independent summands

Proceedings - Mathematical Sciences - A new central limit theorem for independent summands is obtained. In the case of identically distributed summands, it is stronger than the Lévy-Lindeberg...     

A central limit theorem for the U-statistic

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 269–271, February, 1989.     

A central limit theorem for integer partitions

Recently, Hwang proved a central limit theorem for restricted Λ-partitions, where Λ can be any nondecreasing sequence of integers tending to infinity that satisfies certain technical conditions. In particular, one of these conditions is that the associated Dirichlet series has only a single pole on the abscissa of convergence. In the present paper, we show that this condition can be relaxed, and provide some natural examples that arise from the study of integers with restrictions on their digital (base-b) expansion.     

A central limit theorem for empirical histograms

Probability Theory and Related Fields -     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that

A quantum-mechanical functional central limit theorem

Continuing an earlier work [4], properties of canonical Wiener processes are investigated. An analog of the sample path continuity property is obtained. A noncommutative counterpart of weak convergence is formulated. Operator processes (P_n, Q_n) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence (p_i, q_i) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that P_n, Q_n) converges weakly to a canonical Wiener process.     

A central limit theorem for generalized multilinear forms

Let X₁, …, X_n be independent random variables and define for each finite subset I {1, …, n} the σ-algebra = σ{X_i : i ε I}. In this paper -measurable random variables W_I are considered, subject to the centering condition E(W_I ) = 0 a.s. unless I J. A central limit theorem is proven for d-homogeneous sums W(n) = Σ_{I = d}W_I, with var W(n) = 1, where the summation extends over all (_n^d) subsets I {1, …, n} of size I = d, under the condition that the normed fourth moment of W(n) tends to 3. Under some extra conditions the condition is also necessary.     

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form _Xk=g(_εk−s,s∈Z^d)$X_k=g({\epsilon }_{k-s},s\in {\mathbb{Z}}^d)$, k∈Z^d$k\in {\mathbb{Z}}^d$, where _(εi)i∈Zd${\left({\epsilon }_i\right)}_{i\in {\mathbb{Z}}^d}$ are iid random variables and g$g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.     

A tracial quantum central limit theorem

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.

     

A central limit theorem for fuzzy random variables

Fuzzy random variables have been proposed to treat situations in which both random behavior and fuzzy perception must be considered. A definition of independence is given for fuzzy random variables, as well as a notion of fuzzy Gaussian random variables. It is shown that a sum or mean of independent fuzzy random variables converges in the limit to a fuzzy Gaussian random variable, thus providing a fuzzy analogue of the central limit theorem of classical probability theory.     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997