Triangle inequality for Gausisian Random Variable

Question

Ahmad Khalifi 2022-9-27

1
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1812650-triangle-inequality-for-gausisian-random-variable

回答： William Rose 2022-9-27

Triangle inequality must hold for complex-valued Gaussian rando variable! The simulation on matlab shows conflect results (few negatives). kindly see the matlab code below.

clc; clear; clf;

N=10;

R=zeros(N,1); %to record the results

for T=1:N

h=randn(1,2)+1i*randn(1,2); %this is 1x2 vector [v w]

h1=sum(h); %this is a complex number: v+w

R(T,1)= h*h'- h1*h1'; % must be postive based on the triangle inequality

end

plot(R)

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Answer 1

William Rose 2022-9-27

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1812650-triangle-inequality-for-gausisian-random-variable#answer_1061300

在 MATLAB Online 中打开

@Ahmad Khalifi,

N=10;

R1=zeros(N,1); %to record the results

R2=zeros(N,1);

for T=1:N

h=randn(1,2)+1i*randn(1,2); %this is 1x2 vector [v w]

hs=sum(h); %this is a complex number: v+w

R1(T,1)= h*h'- hs*hs'; % must be postive based on the triangle inequality

R2(T,1)=abs(h(1))+abs(h(2))-abs(hs);

end

plot(R1,'-r.'); hold on; plot(R2,'-bx'); legend('R1','R2'); grid on

The thing you are computing, which I have renamed R1, is not the triangle inequality for complex numbers. The quantity R2 is the triangle inequality and it is always positive, as expected.