Sum of iid random variables

Question

Let $X_1, X_2,...,X_n$ be iid random variables. Let $Z_1, Z_2, Z_3$ be defined as $X_1, X_1+X_2, X_1+X_2+X_3$ respectively. Are $Z_1, Z_2$ and $Z_3$ also iid's?

The question is based on renewal processes. If the inter arrival durations are iid, are the arrival times also iid?

Answer 1

Assume that $E[X_i]=0$. If this is not true, then simply subtract the mean, it's a constant, so it will not change anything.

For independent random variables $Cov[Z_nZ_{n-1}]=E[Z_nZ_{n-1}]=0$.

Evaluate the left hand side $$E[Z_nZ_{n-1}] =E[(Z_{n-1}+X_n)Z_{n-1}] =E[X_nZ_{n-1}]+E[Z^2_{n-1}] =E[Z^2_{n-1}]>0 $$ So, $Z_n$ is not independent of $Z_{n-1}$.

Here, we used $E[X_nZ_{n-1}]=0$, because $X_n$ is independent of all $X_i$.

Answer 2

This answer is meant to supplement this excellent analytical answer. I often like to numerically check things to rapidly visualize.

A quick simulation shows this can't be true in general. Let $X_i \stackrel{iid}{\sim} \text{Uniform}(0,1)$.

With 10000 samples, it is clear that $Z_i$ are not i.i.d.

Some MATLAB code below:

% MATLAB R2018a
n = 10000;   % sample size
% Generate samples from Xi ~ U(0,1)
X1 = rand(5000,1);
X2 = rand(5000,1);
X3 = rand(5000,1);

% Get Zi
Z1 = X1;
Z2 = X1 + X2;
Z3 = X1 + X2 + X3;

If two random variables $Y_1$ & $Y_2$ are independent, their correlation will be zero. We know the reverse is not always true (e.g. zero correlation does not imply independence).

% Check correlation for Xi's (should be approx zero)    
% Cx(i,j) = corr(Xi,Xj)
Cx = ones(3);
Cx(1,2) = corr(X1,X2); Cx(2,1) = Cx(1,2);
Cx(1,3) = corr(X1,X3); Cx(3,1) = Cx(1,3);
Cx(2,3) = corr(X2,X3); Cx(3,2) = Cx(2,3);

% Check correlation for Zi's (are they even close to zero?)
Cz = ones(3);
Cz(1,2) = corr(Z1,Z2); Cz(2,1) = Cz(1,2);
Cz(1,3) = corr(Z1,Z3); Cz(3,1) = Cz(1,3);
Cz(2,3) = corr(Z2,Z3); Cz(3,2) = Cz(2,3);

% Visually inspect distributions
figure
s(1)= subplot(1,3,1)
    histogram(Z1,'Normalization','pdf','FaceColor','r')
s(2) = subplot(1,3,2)
    histogram(Z2,'Normalization','pdf','FaceColor','b')
s(3) = subplot(1,3,3)
    histogram(Z3,'Normalization','pdf','FaceColor','g')

% Cosmetics   
ylabel(s(1),'PDF')
for k = 1:3
    xlabel(s(k),['Z' num2str(k)])
    s(k).XLim = [0 4];
    s(k).YLim = [0 1.2];
end