How spurious $t$-statistics grow

Using Proposition 13A.3, show that for two independent random walks the typical size of the OLS slope $t$-statistic grows like $\sqrt T$. (a) In a simulation the median $|t|$ is about 7.1 at $T=250$. What median $|t|$ do you expect at $T = 1{,}000$? Give the ratio. (b) Explain why the median $R^2$ in the simulations of 13A.3 does not fall as $T$ grows. (c) Why does the Durbin–Watson statistic of such a regression tend to zero?