The exact order of normal approximation in bivariate renewal theory

1981 ◽  
Vol 13 (1) ◽  
pp. 113-128 ◽  
Author(s):  
Ibrahim A. Ahmad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem ◽  
Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

The exact order of normal approximation in bivariate renewal theory

1981 ◽  
Vol 13 (01) ◽  
pp. 113-128 ◽  
Author(s):  
Ibrahim A. Ahmad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem ◽  
Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

On the exact order of normal approximation in multivariate renewal theory

1985 ◽  
Vol 22 (02) ◽  
pp. 280-287 ◽  
Author(s):  
Ştefan P. Niculescu ◽  
Edward Omey
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

On the exact order of normal approximation in multivariate renewal theory

1985 ◽  
Vol 22 (2) ◽  
pp. 280-287 ◽  
Author(s):  
Ştefan P. Niculescu ◽  
Edward Omey
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

scholarly journals Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021 ◽  
Author(s):  
Felix Herold ◽  
Daniel Hug ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Hyperbolic Space ◽  
Central Limit ◽  
Hausdorff Measure ◽  
Rates Of Convergence ◽  
Limit Problem ◽  
Dimensional Hausdorff Measure ◽  
The Central Limit Theorem ◽  
Order Quantities

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

First-passage percolation processes with finite height

1985 ◽  
Vol 22 (4) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

scholarly journals An expansion related to the central limit theorem

1969 ◽  
Vol 10 (1-2) ◽  
pp. 219-230
Author(s):  
C. R. Heathcote
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Independent And Identically Distributed

Let X1, X2,…be independent and identically distributed non-lattice random variables with zero, varianceσ2<∞, and partial sums Sn = X1+X2+…+X.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

An invariance principle in k-dimensional extended renewal theory

1979 ◽  
Vol 16 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Covariance Matrix ◽  
Invariance Principle ◽  
Exit Time ◽  
Renewal Theory ◽  
Random Vectors ◽  
First Exit Time ◽  
First Passage ◽  
Passage Times

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

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