In[]:=
Sat 25 Nov 2023 18:49:14
In[]:=
d=1000;getNorm[n_]:=x:=RandomVariate[NormalDistribution[0,],{d,d}];Cn=Nest[x.#&,x,n-1];;TableForm[Table[{n,getNorm[n],getNorm[n],getNorm[n],getNorm[n]},{n,1,5}],TableHeadings->{{},{"n","sample1","sample2","sample3","sample4"}}]
-1/2
d
2
Norm[Cn.Cn,"Frobenius"]
Out[]//TableForm=
n | sample1 | sample2 | sample3 | sample4 | |
1 | 2003.99 | 1998.66 | 1999.51 | 1998.14 | |
2 | 3029.97 | 2990.12 | 3008.21 | 2999.13 | |
3 | 3967.81 | 3995.46 | 4022.33 | 4005.2 | |
4 | 5027.41 | 5075.39 | 4941.94 | 5057.4 | |
5 | 6143.21 | 5964.35 | 5844.76 | 6015.08 |
Moments
Moments
Comparing XX’XX’ vs XXX’X’
Comparing XX’XX’ vs XXX’X’
In[]:=
n=40;numTrials=10;maxMoment=5;sq[mat_]:=mat.mat;randMat:=RandomVariate[NormalDistribution[],{n,n}]
n
;sq[mat_]:=mat.mat;method1:=With[{X=randMat},X.X.X.X];method2:=With[{X=randMat},X.X.X.X];method1f:=With[{X=randMat,Y=randMat},X.X.Y.X];method2f:=With[{X=randMat,Y=randMat},X.X.Y.X];(*momentofamatrix*)moment[mat_,s_]:=Tr[MatrixPower[mat,s]]/n;(*Estimatessmoment*)momentEstimate[s_,method_]:=If[s==0,0,Mean@Table[moment[method,s],{numTrials}]];momentsXY=Table[momentEstimate[s,sq@randMat],{s,1,maxMoment+1}];momentsXYXY=Table[momentEstimate[s,method1],{s,0,maxMoment}];momentsXXYY=Table[momentEstimate[s,method2],{s,0,maxMoment}];momentsXYXYf=Table[momentEstimate[s,method1f],{s,0,maxMoment}];momentsXXYYf=Table[momentEstimate[s,method2f],{s,0,maxMoment}];TableForm[{momentsXY,momentsXYXY,momentsXXYY,momentsXYXYf,momentsXXYYf},TableHeadings->{{"s=1","s=2","s=3"},{"XY","XYXY","XXYY","fXYXY","fXXYY"}}]Out[]//TableForm=
XY | XYXY | XXYY | fXYXY | fXXYY | |
s=1 | 0.97476 | 0 | 0 | 0 | 0 |
s=2 | 1.98833 | 2.13488 | 0.999894 | -0.0203797 | -0.0203088 |
s=3 | 4.87907 | 14.1091 | 3.00952 | 0.168681 | 0.0596858 |
15.19 | 276.574 | 11.4956 | -0.557789 | 0.0810549 | |
53.1827 | 1190.91 | 60.2259 | 1.24756 | 0.211215 | |
102.676 | 23461.2 | 276.462 | 7.67704 | 0.122226 |
In[]:=
momentsXXYYf
Out[]=