# -Mixing Sequence of Random Variables

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### On Complete Convergence for Arrays of Rowwise Mixing Random Variables and Its Applications

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.

### On the Strong Laws for Weighted Sums of Mixing Random Variables

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.

### Complete moment convergence for moving average process generated by $$\rho^{ }$$ mixing random variables

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.

### Some Strong Limit Theorems for Weighted Product Sums of Mixing Sequences of Random Variables

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.

### Almost sure central limit theorems for strongly mixing and associated random variables

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.

### Complete convergence for weighted sums of arrays of rowwise ρ˜ mixing random variables

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. ˜

### Complete Moment Convergence of Weighted Sums for Arrays of Rowwise Mixing Random Variables

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.

### Chover-type laws of the k-iterated logarithm for weighted sums of strongly mixing sequences

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.

### On complete convergence and complete moment convergence for weighted sums of $$\rho^{*}$$ mixing random variables

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.

### Complete moment convergence for product sums of sequence of extended negatively dependent random variables

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.

### PAC Bayesian inequalities of some random variables sequences

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

### Complete moment convergence for randomly weighted sums of martingale differences

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.

### Continuous random variables

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

### Discrete random variables

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.

### On Complex Random Variables

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.

### APPROXIMATION OF RANDOM SUMS OF RANDOM VARIABLES IN INSURANCE

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.

### Some probability inequalities for a class of random variables and their applications

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.

### Uniform convergence of estimator for nonparametric regression with dependent data

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 >

### On the almost sure central limit theorem for self normalized products of partial sums of ϕ mixing random variables

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

### Some conclusions of the exchangeable random variables and the independent identical distribution random variables

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.