scholarly journals A General Central Limit Theorem for Shape Parameters of $m$-ary Tries and PATRICIA Tries

10.37236/3763 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Fuchs ◽  
Chung-Kuei Lee
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Slight Modification ◽  
Wiener Index ◽  
Central Limit Theorems ◽  
Shape Parameters ◽  
Mean And Variance ◽  
The Mean

Tries and PATRICIA tries are fundamental data structures in computer science with numerous applications. In a recent paper, a general framework for obtaining the mean and variance of additive shape parameters of tries and PATRICIA tries under the Bernoulli model was proposed. In this note, we show that a slight modification of this framework yields a central limit theorem for shape parameters, too. This central limit theorem contains many of the previous central limit theorems from the literature and it can be used to prove recent conjectures and derive new results. As an example, we will consider a refinement of the size of tries and PATRICIA tries, namely, the number of nodes of fixed outdegree and obtain (univariate and bivariate) central limit theorems. Moreover, trivariate central limit theorems for size, internal path length and internal Wiener index of tries and PATRICIA tries are derived as well.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (01) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (1) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

scholarly journals CONSTRUCTIVE UNIVERSAL CENTRAL LIMIT THEOREMS BASED ON INTERACTING FOCK SPACES

2005 ◽  
Vol 08 (04) ◽  
pp. 631-650 ◽  
Author(s):  
L. ACCARDI ◽  
V. CRISMALE ◽  
Y. G. LU
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Measure ◽  
Central Limit ◽  
Limit Theorems ◽  
Fock Space ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Fock Spaces ◽  
Interacting Fock Space

Cabana-Duvillard and lonescu11 have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem". Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3 and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.

Functional and random central limit theorems for the Robbins-Munro process

1976 ◽  
Vol 13 (1) ◽  
pp. 148-154 ◽  
Author(s):  
D. L. McLeish
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stopping Rules ◽  
The Central Limit Theorem ◽  
Functional Central Limit

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

scholarly journals On central limit theorems for branching processes with dependent immigration

2020 ◽  
pp. 7-15
Author(s):  
V. Golomoziy ◽  
S. Sharipov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Branching Processes ◽  
Central Limit Theorems ◽  
Standard Normal Distribution ◽  
Immigration Process ◽  
Standard Normal ◽  
Mean And Variance ◽  
The Mean ◽  
Regularly Varying

In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

scholarly journals Large deviations and central limit theorems for sequential and random systems of intermittent maps

2020 ◽  
pp. 1-28
Author(s):  
MATTHEW NICOL ◽  
FELIPE PEREZ PEREIRA ◽  
ANDREW TÖRÖK
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Systems ◽  
Central Limit Theorems ◽  
Random Dynamical Systems ◽  
Stat Phys ◽  
Quenched Central Limit Theorem

Abstract We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

Gordin's theorem and the periodogram

1976 ◽  
Vol 13 (02) ◽  
pp. 365-370 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Stationary Sequence ◽  
Central Limit Theorems ◽  
Stationary Processes

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

Functional and random central limit theorems for the Robbins-Munro process

1976 ◽  
Vol 13 (01) ◽  
pp. 148-154
Author(s):  
D. L. McLeish
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stopping Rules ◽  
The Central Limit Theorem ◽  
Functional Central Limit

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

Sign in / Sign up

Export Citation Format

Share Document