# Limit of random walk auto correlation function

Given the random walk process $$y_{t}=y_{t-1}+e_{t}$$, the auto correlation function is given by $$corr(y_{t}, y_{t-h})=(\frac{t-h}{t})^{1/2}=(1-\frac{h}{t})^{1/2}$$, which tends to 0 as t tends to infinity. What is the intuition behind this result?

Hi: The expression tends to 1.0. The intuition is that, as $$t$$ gets larger and larger, the $$h$$ lags that seperate the two processes become more and more negligible and the processes begin to look the same. This happens because, as $$t$$ gets larger and larger, the proportion of observations common to both processes becomes larger and larger.