scholarly journals Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules

10.37236/4914 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Stavros Kousidis ◽  
Ernst Schulte-Geers
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Current Algebra ◽  
Irreducible Representations ◽  
Moody Algebra ◽  
Asymptotically Normal ◽  
Asymptotic Parameter ◽  
Demazure Modules ◽  
Fusion Products

We prove asymptotic normality of the distributions defined by $q$-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of $\frak{sl}_2$. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to $\frak{sl}_2$. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type $A$ standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, $N$ say.

A central limit theorem for the isotropic random sphere

1995 ◽  
Vol 27 (03) ◽  
pp. 642-651
Author(s):  
Jason J. Brown
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Field ◽  
Random Process ◽  
Positive Constant ◽  
Marginal Distribution ◽  
Dependence Structure ◽  
Sample Mean ◽  
Isotropic Random

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

scholarly journals On the Zagreb Index of Random Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1189-1196 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Process ◽  
Central Limit ◽  
Topological Indices ◽  
Zagreb Index ◽  
Martingale Central Limit Theorem ◽  
Recursive Tree ◽  
Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

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.

scholarly journals A central limit theorem for numbers satisfying a class of triangular arrays associated with Hermite polynomials

2021 ◽  
Vol 61 ◽  
pp. 1-7
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convergence Rate ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Generating Function ◽  
Limit Theorems ◽  
Hermite Polynomials ◽  
Triangular Arrays ◽  
Analytical Expressions

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays. We obtain analytical expressions for the semiexponential generating function the numbers, associated with Hermite polynomials. We apply the results to prove the asymptotic normality of the numbers and specify the convergence rate to the limiting distribution.    

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.

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.

Waitingtimes between record observations

1967 ◽  
Vol 4 (1) ◽  
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 Δr 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 Δr 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.

A central limit theorem for the isotropic random sphere

1995 ◽  
Vol 27 (3) ◽  
pp. 642-651
Author(s):  
Jason J. Brown
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Field ◽  
Random Process ◽  
Positive Constant ◽  
Marginal Distribution ◽  
Dependence Structure ◽  
Sample Mean ◽  
Isotropic Random

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

CENTRAL LIMIT THEOREM FOR THE LOG-REGRESSION WAVELET ESTIMATION OF THE MEMORY PARAMETER IN THE GAUSSIAN SEMI-PARAMETRIC CONTEXT

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 301-313 ◽  
Author(s):  
E. MOULINES ◽  
F. ROUEFF ◽  
MURAD S. TAQQU
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Spectral Density Function ◽  
Dependence Structure ◽  
Asymptotically Normal ◽  
The One ◽  
Gaussian Time

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator [Formula: see text]. We show that the deviation [Formula: see text], adequately normalized, is asymptotically normal and specify the asymptotic variance.

Asymptotic normality of some Graph-Related statistics

1989 ◽  
Vol 26 (1) ◽  
pp. 171-175 ◽  
Author(s):  
Pierre Baldi ◽  
Yosef Rinott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Simple Proof ◽  
Random Function ◽  
Random Variables ◽  
Local Maxima ◽  
Colored Graphs ◽  
U Statistics

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

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