The volume of a random simplex in an n-ball is asymptotically normal

1977 ◽  
Vol 14 (03) ◽  
pp. 647-653 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Asymptotically Normal ◽  
Geometrical Probability ◽  
Mean And Variance ◽  
Random Simplex

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.

The volume of a random simplex in an n-ball is asymptotically normal

1977 ◽  
Vol 14 (3) ◽  
pp. 647-653 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Asymptotically Normal ◽  
Geometrical Probability ◽  
Mean And Variance ◽  
Random Simplex

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.

On a Daley-Kendall model of random rumours

1990 ◽  
Vol 27 (01) ◽  
pp. 14-27 ◽  
Author(s):  
B. Pittel
Keyword(s):  
Asymptotically Normal ◽  
Mean And Variance

Suppose that a certain population consists of N individuals. One member initially learns a rumour from an outside source, and starts telling it to other members, who continue spreading the information. A knower becomes inactive once he encounters somebody already informed. Daley and Kendall, who initiated the study of this model, conjectured that the number of eventual knowers is asymptotically normal with mean and variance linear in N. Our purpose is to confirm this conjecture.

scholarly journals Mean and variance of balanced Pólya urns

2020 ◽  
Vol 52 (4) ◽  
pp. 1224-1248
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Largest Eigenvalue ◽  
Asymptotically Normal ◽  
Pólya Urn ◽  
Polya Urn ◽  
Mean And Variance ◽  
The Mean ◽  
The Largest Eigenvalue

AbstractIt is well-known that in a small Pólya urn, i.e., an urn where the second largest real part of an eigenvalue is at most half the largest eigenvalue, the distribution of the numbers of balls of different colours in the urn is asymptotically normal under weak additional conditions. We consider the balanced case, and then give asymptotics of the mean and the covariance matrix, showing that after appropriate normalization, the mean and covariance matrix converge to the mean and covariance matrix of the limiting normal distribution.

On a Daley-Kendall model of random rumours

1990 ◽  
Vol 27 (1) ◽  
pp. 14-27 ◽  
Author(s):  
B. Pittel
Keyword(s):  
Asymptotically Normal ◽  
Mean And Variance

Suppose that a certain population consists of N individuals. One member initially learns a rumour from an outside source, and starts telling it to other members, who continue spreading the information. A knower becomes inactive once he encounters somebody already informed. Daley and Kendall, who initiated the study of this model, conjectured that the number of eventual knowers is asymptotically normal with mean and variance linear in N. Our purpose is to confirm this conjecture.

TESTING FOR LONG MEMORY IN VOLATILITY

2002 ◽  
Vol 18 (6) ◽  
pp. 1291-1308 ◽  
Author(s):  
Clifford M. Hurvich ◽  
Philippe Soulier
Keyword(s):  
Long Memory ◽  
Null Hypothesis ◽  
Asymptotic Variance ◽  
Wald Test ◽  
Stochastic Volatility Models ◽  
Asymptotically Normal ◽  
Volatility Models ◽  
Regression Estimators ◽  
Mean And Variance ◽  
Short Memory

We consider the asymptotic behavior of log-periodogram regression estimators of the memory parameter in long-memory stochastic volatility models, under the null hypothesis of short memory in volatility. We show that in this situation, if the periodogram is computed from the log squared returns, then the estimator is asymptotically normal, with the same asymptotic mean and variance that would hold if the series were Gaussian. In particular, for the widely used GPH estimator [d with circumflex above]GPH under the null hypothesis, the asymptotic mean of m1/2[d with circumflex above]GPH is zero and the asymptotic variance is π2/24 where m is the number of Fourier frequencies used in the regression. This justifies an ordinary Wald test for long memory in volatility based on the log periodogram of the log squared returns.

Functional limit theorems for critical processes with immigration

2007 ◽  
Vol 39 (4) ◽  
pp. 1054-1069 ◽  
Author(s):  
I. Rahimov
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Asymptotically Normal ◽  
Covariance Functions ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator ◽  
Mean And Variance ◽  
The Mean

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

scholarly journals Functional limit theorems for critical processes with immigration

2007 ◽  
Vol 39 (04) ◽  
pp. 1054-1069 ◽  
Author(s):  
I. Rahimov
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Asymptotically Normal ◽  
Covariance Functions ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator ◽  
Mean And Variance ◽  
The Mean

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

scholarly journals On the number of series parallel and outerplanar graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Manuel Bodirsky ◽  
Omer Gimenez ◽  
Mihyun Kang ◽  
Marc Noy
Keyword(s):  
Expected Value ◽  
Random Series ◽  
Outerplanar Graphs ◽  
Asymptotically Normal ◽  
International Audience ◽  
Mean And Variance

International audience We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g \cdot n^{-5/2} \gamma^n n!$, where $\gamma$ and $g$ are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs.

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