consequences include the Dynkin-Lamperti arcsine laws, see Thaler [@Thaler98]. Indeed our main result significantly simplifies the derivation of the arcsine laws, see [@Thaler00].

Preliminaries
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For convenience, we state Karamata’s Theorem on the integration of regularly varying sequences [@BinghamGoldieTeugels; @Feller66].

\[prop-Karamata\] Suppose that $\ell$ is slowly varying.

-   If $p>-1$, then $\sum_{j=1}^n\ell(j)j^p\sim \frac{1}{p+1}\ell(n)n^{p+1}$ as $n\to\infty$.

-   The function $\tilde\ell(x)
    =\sum_{j=1}^{[x]}\ell(j)j^{-1}$ is slowly varying and $\lim_{n\to\infty}\ell(n)/\tilde\ell(n)=0$. 

The following consequence of (H1) and regularly varying tails is standard.

\[prop-H1\] There is a constant $C>0$ such that $\|R(\rho e^{i(\theta+h)})-R(\rho e^{i\theta})\|\le C m(1/h)h^\beta$ and $\|R(\rho)-R(1)\|\le C m(\frac{1}{1-\rho})(1-\rho)^\beta$ for all $\theta\in[0,2\pi)$, $\rho\in(0,1]$, $h>0$.

We sketch the calculation for the first estimate. By Proposition \[prop-Karamata\], $$R(\rho e^{i(\theta+h)})-R(\rho e^{i\theta}) 
\ll h\sum_{j=1}^Nj\mu(\varphi=j)+\sum_{j>N}\mu(\varphi=j)
\ll hm(N)N^{1-\beta}+m(N)N^{-\beta},$$ so the result follows with $N=[h^{-1}]$.

By (H2), there exists $\epsilon>0$ such that $R(z)$ has a continuous family of simple eigenvalues $\lambda(z)$ for $z\in \bar{{\mathbb D}}\cap B_\epsilon(1)$ with $\lambda(1)=1$. Let $P(z):\mathcal{B}\to\mathcal{B}$ denote the corresponding family of spectral projections, with complementary projections $Q(z)=I-P(z)$ and $P(1)=P$. Also, let $v(z)\in\mathcal{B}$ denote the corresponding family of eigenfunctions normalised so that $\int_Y v(z)\,d\mu=1$ for all $z$. In particular, $v(1)\equiv1$.

\[cor-H1\] The estimates for $R(z)$ in Proposition \[prop-H1\] are inherited by the families $P(z)$, $Q(z)$, $\lambda(z)$ and $v(z)$, where defined.

