scholarly journals Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Dianliang Deng
Keyword(s):  
Convergence Rates ◽  
Broad Class ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Moderate Deviations ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Class Of Functions ◽  
Equivalent Conditions

Let{X,Xn¯;n¯∈Z+d}be a sequence of i.i.d. real-valued random variables, andSn¯=∑k¯≤n¯Xk¯,n¯∈Z+d. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}, wherea(n¯)=n11/α1⋯nd1/αd,b(n¯)=n1β1⋯ndβd,ϕandψare taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).

scholarly journals Some results on convergence rates for probabilities of moderate deviations for sums of random variables

1992 ◽  
Vol 15 (3) ◽  
pp. 481-497 ◽  
Author(s):  
Deli Li ◽  
Xiangchen Wang ◽  
M. Bhaskara Rao
Keyword(s):  
Convergence Rates ◽  
Broad Class ◽  
Random Variables ◽  
Moderate Deviations ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Sums Of Random Variables ◽  
Convergence Of Series ◽  
Class Of Functions ◽  
Equivalent Conditions

LetX,Xn,n≥1be a sequence ofiidreal random variables, andSn=∑k=1nXk,n≥1. Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series∑n≥1(ψ2(n)/n)P(|Sn|≥nφ(n))only under the assumptions convergence thatEX=0andEX2=1, whereφandψare taken from a broad class of functions. These results generalize and improve some recent results of Li (1991) and Gafurov (1982) and some previous work of Davis (1968). Forb∈[0,1]andϵ>0, letλϵ,b=∑n≥3((loglogn)b/n)I(|Sn|≥(2+ϵ)nloglogn).The behaviour ofEλϵ,basϵ↓0is also studied.

scholarly journals Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

2020 ◽  
Vol 22 (4) ◽  
pp. 415-421
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien ◽  
Nguyen Tan Nhut
Keyword(s):  
Limit Theorems ◽  
Negative Binomial ◽  
Convergence Rates ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Weak Limit ◽  
Probability Metric ◽  
Weak Limit Theorems ◽  
Distribution Theorem ◽  
Binomial Sums

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

scholarly journals Maxima of Sums of Heavy-Tailed Random Variables

2002 ◽  
Vol 32 (1) ◽  
pp. 43-55 ◽  
Author(s):  
K.W. Ng ◽  
Q.H. Tang ◽  
H. Yang
Keyword(s):  
Asymptotic Properties ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Large Classes ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Individual ◽  
Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

Application to the Uniform Laws of Large Numbers for Dependent Processes

2019 ◽  
pp. 438-447
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Single Function ◽  
Strongly Mixing ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Dependent Processes

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ‎-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.

Gaussian Approximation under Asymptotic Negative Dependence

2019 ◽  
pp. 277-302
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Gaussian Approximation ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Practical Applications ◽  
Functional Clt ◽  
Dependent Sequence ◽  
Associated Sequences ◽  
Maximal Moment

Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class of random variables is that it leads to the functional CLT without the need to estimate rates of convergence of mixing coefficients. More precisely, because of the maximal moment inequalities established in the previous chapter, we are able to prove tightness for a stochastic process constructed from a negatively dependent sequence. Furthermore, we establish the convergence of the partial sums process, either to a Gaussian process with independent increments or to a diffusion process with deterministic time-varying volatility. We also provide the multivariate form of these functional limit theorems. The results are presented in the non-stationary setting, by imposing Lindeberg’s condition. Finally, we give the stationary form of our results for both asymptotic positively and negatively associated sequences of random variables.

Inequalities for the tail probabilities of weighted sums of independent random variables with applications to rates of convergence to zero

1982 ◽  
Vol 93 (1-2) ◽  
pp. 111-121
Author(s):  
J. E. A. Dunnage
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Special Case

SynopsisWe obtain inequalities for where Wn = anlX1 + … + annXn, the Xr being independent random variables and the Mn being certain truncated means. We then use these inequalities to study the rate at which this probability tends to zero as N→ ∞, noting that in the special case Wn = (X1 + … + Xn)/n, we obtain the estimate given by L. E. Baum and M. Katz which they show is, in a sense, best possible.A desire to find an inequality which would lead to the result of Baum and Katz was, indeed, the impetus behind this paper.

Sign in / Sign up

Export Citation Format

Share Document