$n,N$ limit of $c_{\phi}=\int_{0}^{\infty}\phi(x)dx/2$. If one can show that it is $1/\sqrt{2}$ when $N{\rightarrow}\infty$ through even values, then Johnstone’s considerations hold true all the way and we have the same conclusion as in Theorem \[ThJ\].\
We note that using the interlacing properties of the singular values (as mentioned for instance in @sosh, Remark 5; see also @hj, theorem 7.3.9), as well as the estimates of the difference (resp. ratio) between two consecutive terms of the centering (resp. scaling) sequence, the $N$ odd case follows immediately from the $N$ even case. To be more precise, we use the fact that $$\frac{\mu_{n,N}-\mu_{n,N-1}}{{\sigma_N}}={\mathrm{O}}(N^{-1/3}) \; \; {\rightarrow}0 \text{ as } N {\rightarrow}\infty$$ to check that the $N$ even terms lower and upper bounding the $N$ odd probability have the same limit. Note that the same relationship holds for $\mu_{n+a,N+b}$ and $\mu_{n,N}$, if $a$ and $b$ are fixed real numbers. Therefore, after doing the proof with centering sequence $\mu_{n+3/2,N+1/2}$ (which is technically simpler), we will be able to conclude that the theorem holds true for $\mu_{n,N}$.

Last, to be able to use @sosh, Lemma 2, which gives the result we wish for the joint distribution of the $k$-largest eigenvalues, we will need to verify that 