Center, Scale and Asymptotic Normality for Censored Sums of Independent, Nonidentically Distributed Random Variables

1991 ◽  
pp. 135-177
Author(s):  
Daniel Charles Weiner
Keyword(s):  
Asymptotic Normality ◽  
Random Variables

A central limit theorem for exchangeable variates with geometric applications

1973 ◽  
Vol 10 (4) ◽  
pp. 837-846 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Sum Of Random Variables ◽  
Th Cell ◽  
Occupancy Problems

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

Records from stationary observations subject to a random trend

2015 ◽  
Vol 47 (04) ◽  
pp. 1175-1189 ◽  
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Random Walk ◽  
Asymptotic Normality ◽  
Strong Convergence ◽  
Ergodic Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Convergence Result ◽  
Stationary Sequences ◽  
Stochastic Trend ◽  
Large Numbers

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Y n = X n + T n , n ≥ 1, where (X n ) n ∈ Z is a stationary ergodic sequence of random variables and (T n ) n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

Asymptotic Normality of a Class of Nonparametric Statistics

1987 ◽  
Vol 3 (3) ◽  
pp. 313-347 ◽  
Author(s):  
Munsup Seoh ◽  
Madan L. Puri
Keyword(s):  
Asymptotic Normality ◽  
Order Statistics ◽  
Linear Combination ◽  
Linear Function ◽  
Random Variables ◽  
Weighted Sum ◽  
Rank Statistics ◽  
Concomitants Of Order Statistics ◽  
Signed Rank ◽  
Special Cases

Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

Waitingtimes between record observations

1967 ◽  
Vol 4 (01) ◽  
pp. 206-208 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Continuous Distribution ◽  
Random Variables ◽  
Large Numbers ◽  
Upper Record ◽  
Independent Identically Distributed

If Δ r denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δ r satisfies the weak law of large numbers and a central limit theorem. This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record. Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (02) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of N n , the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

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