# Correlation between logrank (log-rank) test statistics with common control

Say I have a $$K$$ arm experiment that generates survival (time-to-event) endpoints. There are $$K-1$$ experimental arms and a single control arm.

Say I compute a log rank test statistic comparing the hazards for each arm to control. There will be $$K-1$$ standardized test statistics, $$Z_1, \dots, Z_{k-1}$$. What is the correlation between test statistics?

I've found course-notes that say when $$K=2$$ the correlation is 0.5. Under the null they would be distributed as $$(Z_1, Z_2) \sim \mathcal{N}_2(0, \Sigma)$$, where $$\Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}.$$ This seems plausible but I don’t see why it’s true. If it is true does it expand to multiple log rank test statistics?