Pearson's correlation coefficient and Wiener processes

Question

I have two Wiener processes (Brownian motion), W1 and W2. The relation between them is given by W2= W1 + 0.002*t where t is time index taking values from 1 to 500. If you plot these two processes, you will see that they are diverging away like below:

However, their Pearson's correlation coefficient is quite high and equal to 0.92. I don't understand why for two diverging series like this, Pearson's correlation coefficient would produce such a high value. In other words, why can't Pearson's correlation identify their divergence? Can anyone please explain the theoretical reason behind this? Thanks in advance.

Matlab codes for reproducing these graphs:

%Creating two Wiener processes

randn('state',100)          % set the state of randn
T = 1; N = 500; dt = T/N;

t=1:N;

dW = sqrt(dt)*randn(1,N);   % increments
W1 = cumsum(dW);             % cumulative sum



W2= W1+.002*t;           %Create the second Wiener process

c=num2str(corr(W1',W2')) %Computing correlation and converting it to string
                         %to use it later in the plot annotation box

figure()
plot([0:dt:T],[0,W1],'k-')   % plot W1 against t
xlabel('t','FontSize',10)
ylabel({'W_1(t)';' W_2(t)'},'FontSize',10,'Rotation',0)

hold on


plot([0:dt:T],[0,W2],'b-.')   % plot W2 against t

legend('W_1','W_2')
dim = [0.15 0.45 0.3 0.3];
str = 'Correlation=';
annotation('textbox',dim,'String', {str;c},'FitBoxToText','on');

Answer 1

There are a lot of ways to answer this question, but the bottom line is that the Sample correlation is not well-suited for measuring the "divergence" of two things. Case in point, say $x_1(t) = \frac{1}{2}t$ and $x_2(t) = t.$ These certainly diverge, but you should be able to see that the correlation between them is one.

In general, using the sample correlation (and simple regression analysis and many other sample statistics that are bread and butter in 'iid' statistics) for any purpose is a bad idea when you have non-stationary and trending series.