For arrays of rowwise independent **random** **variables**, complete convergence has been extensively investigated see, e.g., Hu et al. 13, Sung et al. 14, and Kruglov et al. 15. Recently, complete convergence for arrays of rowwise dependent **random** **variables** has been considered. We refer to Kuczmaszewska 16 for ρ-**mixing** and ρ-**mixing** sequences, Kuczmaszewska 17 for negatively associated **sequence**, and Baek and Park 18 for negatively dependent **sequence**. In the paper, we study the complete convergence for arrays of rowwise ρ-**mixing** **sequence** under some suitable conditions using the techniques of Kuczmaszewska 16, 17. We consider the case of complete convergence of maximum weighted sums, which is diﬀerent from Kuczmaszewska 16. Some results also generalize some previous known results for rowwise independent **random** **variables**.

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Hsu and Robbins 12 and Erd ¨os 13 proved the case r 2 and p 1 of the above theorem. The case r 1 and p 1 of the above theorem was proved by Spitzer 14. An and Yuan 9 studied the weighted sums of identically distributed ρ ∗ -**mixing** **sequence** and have the following results.

Obviously, ρ – -**mixing** **random** **variables** include NA and ρ ∗ -**mixing** **random** **variables**, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained; we refer to Wang and Lu [] for a Rosenthal-type moment inequality and weak convergence, Budsaba et al. [, ] for complete conver- gence for moving average process based on a ρ – -**mixing** **sequence**, Tan et al. [] for the almost sure central limit theorem. But there are few results on the complete moment con- vergence of moving average process based on a ρ – -**mixing** **sequence**. Therefore, in this paper, we establish some results on the complete moment convergence for maximum par- tial sums with less restrictions. Throughout the sequel, C represents a positive constant although its value may change from one appearance to the next, I { A } denotes the indicator function of the set A.

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In particular, taking m 1, the above formula is the well-known Marcinkiewicz strong law of large numbers. Thus, our Theorem 2.5 and Corollary 2.6 generalize and improve the Marcinkiewicz strong law of large numbers from the i.i.d. case to ρ-**mixing** **sequence**. In addition, by Theorem 4 in Wu and Jiang 7 is a special case of Corollary 2.6.

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We prove an almost sure central limit theorem (ASCLT) for strongly **mixing** **sequence** of **random** **variables** with a slightly slow **mixing** rate α(n) = O((log logn) − 1 − δ ). We also show that ASCLT holds for an associated **sequence** of **random** **variables** without a stationarity assumption.

Hsu and Robbins [] proved that the **sequence** of arithmetic means of independent and identically distributed (i.i.d.) **random** **variables** converges completely to the expected value if the variance of the summands is ﬁnite. Erdös [] proved the converse. The result of Hsu- Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. See, for example, Spitzer [], Baum and Katz [], Gut [], Zarei [], and so forth. The main purpose of the paper is to provide complete convergence for weighted sums of arrays of rowwise ρ-**mixing** **random** **variables**. ˜ Firstly, let us recall the deﬁnitions of sequences of ρ-**mixing** **random** **variables** and arrays ˜ of rowwise ρ-**mixing** **random** **variables**. ˜

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The main purpose of this paper is to discuss again the above results for arrays of rowwise ϕ-**mixing** **random** **variables**. The author takes the inspiration in 8 and discusses the complete moment convergence of weighted sums for arrays of rowwise ϕ-**mixing** **random** **variables** by applying truncation methods. The results of Ahmed et al. 8 are extended to ϕ- **mixing** case. As an application, the corresponding results of moving average processes based on a ϕ-**mixing** **random** **sequence** are obtained, which extend and improve the result of Kim and Ko 9.

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In this contribution, we derive the Law of the k -Iterated Logarithm for weighted sums of strongly **mixing** sequences of **random** **variables** with infinite second moment. In this respect, the subject of this note belongs in the family of the so-called “Chover-type LIL” - see e.g. Chover [3], Mikosch [6], Vasudeva [12], Cai [1][2], Wu and Jiang [14][15]. One major difference with such contributions, however, is that in our case the upper and the lower halves of the LIL are separated. In particular, the upper half of the LIL is exactly the same as in other studies. On the other hand, the lower half is substantially different: a different norming **sequence** is required, which is slower the slower the convergence of α (m) to zero as m → ∞. As a consequence, the LIL bifurcates into two separate results, thereby not being a sharp result any more.

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In this paper, we show that (1.4) holds for a **sequence** of identically distributed ρ ∗ -**mixing** **random** **variables** with suitable moment conditions. The moment conditions for the cases α < rp and α > rp are optimal. The moment conditions for α = rp are nearly optimal. Al- though the main tool is the Rosenthal moment inequality for ρ ∗ -**mixing** **random** **variables**, our method is simpler than that of Sung [1] even in the case α > rp.

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The aim of this paper is to extend and improve Chow’s result in i.i.d. case to extended negatively dependent (END) **random** **variables**. Some new suﬃcient conditions of com- plete moment convergence results for product sums of **sequence** of extended negatively dependent (END) **random** **variables** are obtained.

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Remark . Hitczenko [] proved the above inequality for conditionally symmetric mar- tingale diﬀerence sequences, and De la Peña [] obtained the above inequality without the martingale diﬀerence assumption, hence without any integrability assumptions. Note that any **sequence** of real valued **random** **variables** X i can be ‘symmetrized’ to produce an

Robbins [] obtained that the **sequence** of arithmetic means of independent and identi- cally distributed (i.i.d.) **random** **variables** converges completely to the expected value if the variance of the summands is ﬁnite. Erdös [] proved the converse. The result of Hsu- Robbins-Erdös is a fundamental theorem in probability theory, and it has been generalized and extended in several directions by many authors. Baum and Katz [] gave the following generalization to establish a rate of convergence in the sense of Marcinkiewicz-Zygmund- type strong law of large numbers.

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What is the probability that X = 28 exactly ? As there is an infinite number of real numbers in the interval [10, 60], if c is any number in this interval, Pr( X = c ) must be 0, otherwise the sum of all the probabilities would be more than 1. This is a very important property of a continuous **random** variable X : for any possible value x , Pr( X = x ) = 0. It only makes sense to consider the probability that X lies in an interval. Thus, although Pr( X = 28) = 0, the

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A **random** variable (denoted by X , Y , …) is a variable whose value is determined by the outcome of a **random** experiment. For example, if an experiment consisted of tossing two standard dice, the **random** variable X could be defined as the number of 6s obtained. In this case, X could have the value 0, 1 or 2. Then X = 2 would denote the event of rolling two 6s; X ≥ 1 the event of rolling at least one 6, and so on. As these are events, we can consider their probabilities, such as Pr(X = 2), i.e. the probability of rolling two 6s.

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In this article we have shown that complex multivariate **random** variable of Z is a complex multivariate normal variable of dimension n if all nondegenerate complex linear combination of Z have a complex univariate normal distribution. The characteristic function of Z has been derived. An extension of complex multivariate t-distribution has been proposed and few examples are suggested.

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ABSTRACT. The paper deals with approximations of **random** sums. By **random** sum we mean sum of **random** number of independent and identically distributed **random** **variables**. Distribution of this sum is called compound distribution. The model is especially important in non-life insurance. There are many methods for approximating compound distributions, one of most popular is approximation with shifted gamma distribution. In this work we show alternative way – using kernel density, Fast Fourier Transform and numerical optimization methods – for finding shifted gamma approximations and show results suggesting its superiority over classical method.

The paper is organized as follows. The exponential probability inequalities for a se- quence of acceptable **random** **variables** are presented in Section , and the complete con- vergence for it is obtained in Section . Our results are based on some moment conditions, while the main results of Sung et al. [] are under the condition of moment and identical distribution.

The parametric θ in Theorem . and Theorem . plays the role of a bridge between the process (i.e. **mixing** exponent) and choice of positive bandwidth h. For example, if d = , θ = , and β > max { s+ s– , s– s– }, then we take h = n –/ in Theorem . and obtain the convergence rate | m n (x) – m(x)| = O P ((ln n) / n –/ ). Similarly, if d = , θ = , and β >

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We ﬁrst prove that (.) holds under condition (.). Note that E|X| p < ∞ for all < p < since X belongs to the domain of attraction of the normal law. For our purpose, we ﬁx / < p < . By the Marcinkiewicz-Zygmund strong law of a large number for φ -**mixing** sequences (see [, Remark ..], []), for i large enough, we have

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As the fundamental structure theorem of infinite exchangeable **random** **variables** sequences, the Definetti’s theorem states that infinite exchangeable **random** **variables** sequences is independent and identically distributed with the condition of the tail σ-algebra. So some results about independent identically distributed **random** **variables** is similar to exchangeable **random** **variables**. As the fundamental structure theorem of infinite exchangeable **random** **variables** sequences, the Definetti’s theorem does not work to finite exchangeable **random** **variables** sequences, it is therefore necessary to find other techniques to solve the approximate behavior problems of finite exchangeable **random** **variables** sequences. By using reverse martingale approach, some scholars have given some results. In this paper we do some researches about the similarity and difference of identically distributed **random** **variables** and exchangeable **random** **variables** sequences, mainly discuss the limit theory of exchangeable **random** **variables**.