Random Vectors and the Multivariate Normal Distribution

2022 ◽  
pp. 37-60
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Multivariate Normal ◽  
Random Vectors

scholarly journals Calculations Involving the Multivariate Normal and Multivariate t Distributions with and without Truncation

2018 ◽  
Vol 18 (4) ◽  
pp. 826-843
Author(s):  
Michael J. Grayling ◽  
Adrian P. Mander
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Distribution Functions ◽  
Multivariate Normal ◽  
Random Vectors ◽  
Multivariate T Distribution ◽  
T Distribution ◽  
Multivariate T

In this article, we present a set of commands and Mata functions to evaluate different distributional quantities of the multivariate normal distribution and a particular type of noncentral multivariate t distribution. Specifically, their densities, distribution functions, equicoordinate quantiles, and pseudo–random vectors can be computed efficiently, in either the absence or the presence of variable truncation.

scholarly journals Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

10.37236/6374 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Cecilia Holmgren ◽  
Svante Janson ◽  
Matas Sileikis
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Normal Limit ◽  
Preferential Attachment ◽  
Multivariate Normal ◽  
Search Trees ◽  
Random Vectors ◽  
Limit Laws ◽  
Fixed Tree

We study fringe subtrees of random $m$-ary search trees and of preferential attachment trees, by putting them in the context of generalised Pólya urns. In particular we show that for the random $m$-ary search trees with $ m\leq 26 $ and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree $ T $ converges to a normal distribution; more generally, we also prove multivariate normal distribution results for random vectors of such numbers for different fringe subtrees. Furthermore, we show that the number of protected nodes in random $m$-ary search trees for $m\leq 26$ has asymptotically a normal distribution.

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