Abstract:
Let $(X_i,i\geq 1)$ be a sequence of i.i.d. random variables with values in $[0,1]$, and $f$ be a function such that $`E(f(X_1)^2)<+\infty$. We show a functional central limit theorem for the process $t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq t}$.

Abstract:
Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process. 1. Introduction Let be a sequence of random variables and define the partial sum and for , where . In the past years, the asymptotic behaviors of the products of various random variables have been widely studied. Arnold and Villase？or [1] considered sums of records and obtained the following form of the central limit theorem (CLT) for independent and identically distributed (i.i.d.) exponential random variables with the mean equal to one, Here and in the sequel, is a standard normal random variable, and ( ) stands for convergence in distribution (in probability, almost surely). Observe that, via the Stirling formula, the relation (1.1) can be equivalently stated as In particular, Rempa？a and Weso？owski [2] removed the condition that the distribution is exponential and showed the asymptotic behavior of products of partial sums holds for any sequence of i.i.d. positive random variables. Namely, they proved the following theorem. Theorem A. Let be a sequence of i.i.d. positive square integrable random variables with and the coefficient of variation . Then, one has Recently, the above result was extended by Qi [3], who showed that whenever is in the domain of attraction of a stable law with index , there exists a numerical sequence (for , it can be taken as ) such that as , where . Furthermore, Zhang and Huang [4] extended Theorem A to the invariance principle. In this paper, we aim to study the weak invariance principle for self-normalized products of trimmed sums of i.i.d. sequences. Before stating our main results, we need to introduce some necessary notions. Let be a sequence of i.i.d. random variables with a continuous distribution function . Assume that the right extremity of satisfies and the limiting tail quotient exists, where . Then, the above limit is for some , and or is said to have a thick tail if , a medium tail if , and a thin tail if . Denote . For a fixed constant , we say is a near-maximum if and only if , and the number of near-maxima is These concepts were first introduced by Pakes and Steutel [5], and their limit properties have been widely studied by Pakes and Steutel [5], Pakes and Li [6], Li [7], Pakes [8], and Hu and Su [9]. Now, set where which are the sum of near-maxima and the trimmed sum, respectively. From Remark 1 of Hu

Abstract:
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 [math.CO/9906120,math.PR/9909104].

Abstract:
We prove a variant of the central limit theorem (CLT) for a sequence of i.i.d. random variables $\xi_j$, perturbed by a stochastic sequence of linear transformations $A_j$, representing the model uncertainty. The limit, corresponding to a "worst" sequence $A_j$, is expressed in terms of the viscosity solution of the $G$-heat equation. In the context of the CLT under sublinear expectations this nonlinear parabolic equation appeared previously in the papers of S.Peng. Our proof is based on the technique of half-relaxed limits from the theory of approximation schemes for fully nonlinear partial differential equations.

Abstract:
Applying the moment inequality of negatively dependent random variables which was obtained by Asadian et al. (2006), the strong limit theorem for weighted sums of sequences of negatively dependent random variables is discussed. As a result, the strong limit theorem for negatively dependent sequences of random variables is extended. Our results extend and improve the corresponding results of Bai and Cheng (2000) from the i.i.d. case to ND sequences.

Abstract:
By means of the notion of likelihood ratio, the limit properties of the sequences of arbitrary-dependent continuous random variables are studied, and a kind of strong limit theorems represented by inequalities with random bounds for functions of continuous random variables is established. The Shannon-McMillan theorem is extended to the case of arbitrary continuous information sources. In the proof, an analytic technique, the tools of Laplace transform, and moment generating functions to study the strong limit theorems are applied.

Abstract:
We prove a central limit theorem for random sums of the form $\sum_{i=1}^{N_n} X_i$, where $\{X_i\}_{i \geq 1}$ is a stationary $m-$dependent process and $N_n$ is a random index independent of $\{X_i\}_{i\geq 1}$. Our proof is a generalization of Chen and Shao's result for i.i.d. case and consequently we recover their result. Also a variation of a recent result of Shang on $m-$dependent sequences is obtained as a corollary. Examples on moving averages and descent processes are provided, and possible applications on non-parametric statistics are discussed.

Abstract:
We prove entropic and total variation versions of the Erd\H{o}s-Kac limit theorem for the maximum of the partial sums of i.i.d. random variables with densities.

Abstract:
We study the weak convergence in the space of processes constructed from products of sums of independent but not necessarily identically distributed random variables. The presented results extend and generalize limit theorems known so far for i.i.d. sequences. 1. Introduction Let be a sequence of independent, positive, and square-integrable random variables (r.v.’s) defined on some probability space . For every let us put Arnold and Villase？or [1] obtained the central limit theorem for sums of records. Rempa？a and Weso？owski [2] observed that this result was of general nature and proved that for a sequence of i.i.d. positive and square-integrable r.v.’s one has where is a standard normal variable and and are the common mean and the coefficient of variation of the r.v.’s. The convergence in (2) was generalized and extended by many authors in different ways. For example, Zhang and Huang [3] studied the functional version of this result. They proved under general conditions, not involving the dependence structure of the sequence, that here and in the sequel denotes the Wiener process. The functional version of the convergence of products of sums was also studied by Kosiński [4], for i.i.d. sequences belonging to the domain of attraction of the -stable law with . The first result for sequences of nonidentically distributed r.v.’s was obtained by Matu？a and St？pień [5]. Similarly as in the invariance principle, the study of the non-i.i.d. sequences requires defining the processes in a different way than in (3). For and each let us define the function , where . Let us introduce the process and for such that , where . In [5] it was proved that for a sequence of independent, positive, and square-integrable random variables satisfying the Lindeberg condition and such that , the process converges weakly in the space to the process . This result was proved in one theorem for the function and in the second for functions . Our goal is to prove the aforementioned result for a large class of continuous functions on , which may be unbounded. Furthermore, we shall slightly change the process in order to avoid its artificial definition in the case and replace the strange shifted index in . In this setting, the imposed conditions and obtained results will be much more natural. 2. Main Results Let be the family of functions which are continuously differentiable on that is, , furthermore, such that Let us observe that from (5) it follows that is integrable, if this function is also monotonic then, in consequence, (6) is satisfied. It is obvious that . The family also