Kljuˇ cne rijeˇ ci: Matematiˇ cko oˇ cekivanje, sluˇ cajna varijabla, mjera, vjerojatnost, niz, ko- nvergencija, red, nejednakost, nezavisnost, konstanta
Abstract: In this paper we will talk about conditions in which sequence of random variables converge, convergence rate and convergence probability. At the beginning we will recall some of the basic terms of probability such as independence of random variables, mathematic expectation and variance and Cebˆıs , ev inequality. We will define few types of convergence of random variables, introduce the term of measure and measure integration and, comparing with them, we will state and prove Cebˆıs , ev, Bernoulli and Khinchin weak **law** theorem of **large** **numbers** by using cutting method. We will be analyzing sequence of independent random variables that have ”zero – one” characteristic, Borel and Kolmogorov theorem and introduce term of tail functions and tail events. Furthermore, we will define empirical function of distribution, study convergence of series of random variables and prove Borel, Kolmogorov, Chung and Cantelli strong **law** of **large** **numbers**. In the end, we will bring up basic theorems needed to prove the **law** of repeated logarithm.

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Zygmund theorem for pairwise NQD random variables.
1 Introduction
Let {X n , n ≥ 1} be a sequence of independent and identically dis- tributed random variables. There are two famous theorems on the strong **law** of **large** **numbers** for such a sequence: Kolmogorov’s the- orem and Marcinkiewicz-Zygmund theorem (see a.g. Loeve, 1963 or Stout, 1974). In what follows, we put S n = P n k=1 X k . Accord- ing to Kolmogorov’s theorem, there exists a constant b such that

Section 7C: The **Law** of **Large** **Numbers**
Example. You flip a coin 100 times.
I Suppose the coin is fair. How many times would you expect to get heads? tails?
One would expect a fair coin to come up heads half the time and tails the other half of the time. So if a coin is being flipped 100 times, one could expect about 1 2 × 100 = 50 heads and about 1 2 × 100 = 50 tails.

JEL classification: C60 · C70
Keywords: Exact **law** of **large** **numbers** · Fubini extension · Incomplete informa- tion · Purification · Saturated probability space
∗ The authors are very grateful to Yi-Chun Chen, Sengkee Chua, Haifeng Fu, M. Ali Khan, Xiao Luo and Yeneng Sun for helpful discussions. The first draft of this paper was prepared when Yongchao Zhang participated the Trimester Program on Mechanism Design held at the Hausdorff Research Institute for Mathematics (HIM), Bonn University, May-August 2009. Financial support from HIM is specially acknowledged.

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An extensive literature in economics uses a continuum of random variables to model in- dividual random shocks imposed on a **large** population. Let H denote the Hilbert space of square-integrable random variables. A key concern is to characterize the family of all H-valued functions that satisfy the **law** of **large** **numbers** when a **large** sample of agents is drawn at random. We use the iterative extension of an infinite product measure introduced in [6] to formulate a “sharp” **law** of **large** **numbers**. We prove that an H-valued function satisfies this **law** if and only if it is both Pettis-integrable and norm integrably bounded.

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Received September 3, 2012; revised October 3, 2012; accepted October 10, 2012
ABSTRACT
Strong **law** of **large** **numbers** is a fundamental theory in probability and statistics. When the measure tool is nonadditive, this **law** is very different from additive case. In 2010 Chen investigated the strong **law** of **large** **numbers** under upper probability V by assuming V is continuous. This assumption is very strong. Upper probabilities may not be continuous.

We discuss strong **law** of **large** **numbers** and complete convergence for sums of uniformly bounded negatively associate NA random variables RVs. We extend and generalize some recent results.
As corollaries, we investigate limit behavior of some other dependent random sequence.
1. Introduction

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Abstract
We consider a model of correlated defaults in which the default times of multiple entities depend not only on common and specific factors, but also on the extent of past defaults in the market, via the average loss process, including the average number of defaults as a special case. The paper characterizes the average loss process when the number of entities becomes **large**, showing that under some monotonicity conditions the limiting average loss process can be determined by a fixed point problem. We also show that the **Law** of **Large** **Numbers** holds under certain compatibility conditions.

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Copyright q 2010 Wang Xuejun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We establish some maximal inequalities for demimartingales which generalize the result of Wang 2004. The maximal inequality for demimartingales is used as a key inequality to establish other results including Doob’s type maximal inequality, strong **law** of **large** **numbers**, strong growth rate, and integrability of supremum for demimartingales, which generalize and improve partial results of Christofides 2000 and Prakasa Rao 2007 .

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ANNA KUCZMASZEWSKA and DOMINIK SZYNAL (Received 4 August 1998 and in revised form 26 March 1999)
Abstract. We give Chung-Teicher type conditions for the SLLN in general Banach spaces under the assumption that the weak **law** of **large** **numbers** holds. An example is provided showing that these conditions can hold when some earlier known conditions fail.

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A **LAW** OF **LARGE** **NUMBERS** FOR FINITE-RANGE DEPENDENT RANDOM MATRICES
GREG ANDERSON AND OFER ZEITOUNI
Abstract. We consider random hermitian matrices in which distant above- diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings.

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makes the (unrealistic) assumption that gains between different pairs of antennas are uncorrelated. The models studied in this work would allow correlation between neighboring antenna pairs. We do not develop this application further here.
The structure of the paper is as follows. In the next section we describe the class of matrices we treat, and state our main results, namely Theorem 2.3 (assert- ing a **law** of **large** **numbers**) and Theorem 2.4 (asserting algebraicity of a Stieltjes transform). We also prove Theorem 2.5 (which is essentially folkloric and explains the regularity implied by algebraicity). In Section 3 we discuss the limit measure in detail, and in particular write down algebro-integral equations for its Stieltjes transform, which we call color equations. Section 4 provides a computation of limits of traces of powers of the matrices under consideration. In Section 5 we complete the proof of Theorem 2.3 by a variance computation. In Section 6 we set up the algebraic machinery needed to prove Theorem 2.4. We finish proving the theorem in Section 7 by analyzing the color equations.

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Considering a sequence of independent and identically distributed (IID) replications of a gamble that is initially rejected, Samuelson (1963) famously demonstrates the “fallacy of **large** **numbers**” in a situation involving the summation of risks. As Ross (1999, p. 324) remarks, “Samuelson points out that the **law** of **large** **numbers** applies to averages and not to sums”. Samuelson (1963) also notes that a situation that involves summing risks corresponds to an insurance company pooling many risks together. Finally, Samuelson (1963) emphasizes that it is not sufficient to merely consider the probability of loss (which is typically decreasing when the number of IID risks increases). Instead, it is the expected utility of wealth that is central to the evaluation.

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In this section, we shall establish the strong **law** of **large** **numbers** for countable nonhomogeneous Markov chains under the condition of uniform convergence in the Cesàro sense. The results obtained are then applied to the Shannon–McMillan–Breiman theorem for Markov chains.
Theorem 1. Let {Xn, n 0 } be a nonhomogeneous Markov chain taking values in the state space S = {1, 2, . . .} with the transition matrices {Pn = (pn (i, j)), n 1 }, and let {fn (x, y), n 1 } be a sequence of func-

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The weak **law** of **large** **numbers** (WLLN) proved by Kolmogorov is one of the most beautiful results in classical probability theory. Quite recently, this funda- mental limit theorem was extended by Bercovici and Pata (1996) to the general context of noncommutative probability theory. The later context means replac- ing the classical probability space (Ω, F, P ) by a noncommutative probability

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n → ∞
The question underlying the present work is how one may refine Theorem CG1 to give more information on the **law** of { X n } . We recall the classical answer, the strong **law** of **large** **numbers** Baum and Katz [1] for (see [2]). In Section 3 we generalize Theorem CG1 and give Baum-Katz’s [1] type results on estimate for the rate of convergence in these laws.

China
Abstract
In this paper, we study the strong **law** of **large** **numbers** for a class of random variables satisfying the maximal moment inequality with exponent 2. Our results embrace the Kolmogorov strong **law** of **large** **numbers** and the Marcinkiewicz strong **law** of **large** **numbers** for this class of random variables. In addition, strong growth rate for weighted sums of this class of random variables is presented.

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= E X 2 + E Y 2
(since E[X] = E[Y ] = 0)
= var(X) + var(Y )
The **Law** of **Large** **numbers** Suppose we perform an experiment and a measurement encoded in the random variable X and that we repeat this experiment n times each time in the same conditions and each time independently of each other. We thus obtain n independent copies of the random variable X which we denote

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Chapter 7
Discussion
The strong **law** of **large** **numbers** for the semiparametric U-statistics estimator S se 2,n , under proper conditions, has been established in Theorem 1.4 . In addition to the assumptions made in Dikta ( 2000 ) and Bose and Sen ( 1999 ), we assumed that the censoring model, i. e. conditional expectation of the censoring indicator given the observation, is a monotone non-decreasing function. However Chapter 5 shows a variety of examples, which are relevant in the field of survival analysis, for which this additional condition is satisfied. These examples include, among others, the proportional hazards model. The product limit estimator, upon which the semi- parametric U-Statistics is based in this example, has the same asymptotic proper- ties as the Cheng and Lin ( 1987 ) estimator (c. f. Dikta ( 2000 ), page 3). In Chapter 6 , we conducted simulation studies for different scenarios. The simulation studies verify the SLLN result in Theorem 1.4 . Moreover the studies show that the semi- parametric estimator outperforms the Kaplan-Meier estimate, especially in terms of variance, in most cases. This was expected because of the results established by Dikta et al. ( 2005 ) and Dikta ( 2014 ). The gain in efficiency was especially **large** for smaller sample sizes. The results of Section 6.4 indicate, that the semiparametric estimator might still be consistent, even if the censoring model is not non-decreasing.

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We prove a **law** of **large** **numbers** for a class of Z d -valued random walks in dynamic random environ- ments, including non-elliptic examples. We assume for the random environment a mixing property called conditional cone-mixing and that the random walk tends to stay inside wide enough space–time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni (2004) [5]

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