3.3 Asymptotic development
Is it possible to compute this p-value faster ? In the case where R admits a diagonal form, simple linear algebra could help to cut off the computations and answer yes to this question.
Proposition 4. If R admits a diagonal form we have
where []1 denotes the first component of a vector, with R∞ = limi→∞ Ri/λi, where 0 <λ < 1 is the largest eigenvalue of R and ν is the magnitude of the second largest eigenvalue. We also have v = [g(a),...,g(1)]'.
Proof. By using the corollary 15 (appendix A) we know that
Ri - λiR∞ = O(νi)     (15)
uniformly in i so we finally get for all α
uniformly for all n ≥ α and the proposition is then proved by considering the first component of equation (16).     □
Corollary 5. We have
and
Proof. Simply replace the terms in (17) and (18) with equation (14) to get the results.     □