Variance Laplacian: Quadratic Forms in Statistics

EasyChair Preprint no. 821, version history

VersionDatePagesVersion notes
1March 12, 201912
2April 5, 201916

(1)  Bounding  all  eigenvalues  using  Lemma  4

(2)  Spectral  Representation  of  Variance  Laplacian  matrix

(3) Probabilistic  interpretation  of   Tsallis  Entropy

3June 11, 201920

(1)  Unit  Random  processes  ( and  the  variance  values )  are  related  to  unit  random  processes

(2)  New  results  on  the  Laplacian  matrix  associated  with  Variance  of  a  discrete  random  variable  are  included

4July 6, 201923

(1)  Characterization  of  Entropic  Quadratic  Forms (  satisfying  reasonable  axioms )  is  discussed

(2)  Relationship  between  Tsallis  Entropy  and  Renyii  entropy  is  discussed.

5August 9, 201928

(1)  Relationship  between  Renyi  entropy  and  Tsallis  entropy  is  derived

(2)   Lemma  5  (  new  lemma )  is  proved  and  its  probabilistic  interpretation  is  provided

(3)  Generalized  Verhulst  maps  are  discussed

6September 5, 201933

(1)  Matrix  Logistic  Map  and  the  associaited  generalized  Verhulst  dynamical  system  is  defined  and  the  dynamic  behavior  is  studied

(2) Lemma  7  is  generalized

(3)  Minor  mistakes  are  corrected

7September 20, 201936

(1)  Lemma  8  is  proved

(2)  Existing  inequality is  reasoned  to  be  tighter  than  Cauchy-Schwarz  inequality.   

(3)  Jensen  inequality  based  inequalities  are  discussed.

8November 1, 201938

(1)  Entire   New  Section  ( i.e.  Section  5 )  is  added.  It  deals  with  approximation  of  Gibbs- Shannon  Entropy  and  the  associated  Information  theoretic  results.

(2)  Mutual  Information  based  on  approximations  is  computed

9November 16, 202140

Covariance  matrix  of   two  random  variables  which  assume  same  values  is   proved  to  be Laplacian.

The  leading  diagonal  sums  of  variance  Laplacian   matrix  are  proved  to  be  the  autocorrelation  coefficients  of  PMF  vector

Eigenvalues  of  Variance  Laplacian  matrix  are   bounded  in  a  tighter  manner

Keyphrases: eigenvalues, Eigenvectors, Laplacian matrix, quadratic form, variance

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
  author = {Rama Murthy Garimella},
  title = {Variance  Laplacian:  Quadratic  Forms  in  Statistics},
  howpublished = {EasyChair Preprint no. 821},

  year = {EasyChair, 2021}}