Correlation coefficient of the least terms of variational series (Q2719509)

Let \((X_i,Y_i)\), \( i=1,2,\dots\), be a sequence of independent random vectors distributed by a similar law according to the density \(f(x,y)\): NEWLINE\[NEWLINEf(x,y)>0\quad\forall (x,y)\in D,\quad f(x,y)\equiv 0\quad\forall (x,y)\not\in D. NEWLINE\]NEWLINE For the main asymptotics \(\rho:=\lim\limits_{n\to\infty} r_n\), where \( r_n:=\mathcal E_{\text{cor}}(\xi_n,\eta_n) \) is the correlation coefficient between \( \xi_n:=\min\limits_{1\leq i\leq n} X_i \) and \( \eta_n:=\min\limits_{1\leq i\leq n} Y_i\), the formula NEWLINE\[NEWLINE\rho=(4-\pi\sqrt{\theta}+2\sqrt{\theta-1} \arctan \sqrt{\theta-1})[(4-\pi)\sqrt{\theta}]^{-1}NEWLINE\]NEWLINE is established.