Let $\mathcal{T}_n$ denote a subcritical Galton-Watson tree conditioned on having $n$ vertices. Assume that the offspring distribution $\xi$ is heavy-tailed and that there is an integer $k_0$ with $$ \mathbb{P}(\xi =k) >0 \quad \text{for all }k \ge k_0. $$ It is known that the degree $d^+_{\mathcal{T}_n}(o)$ of the root vertex $o$ converges in distribution to a random variable $\hat{\xi}$ with values in $\bar{\mathbb{N}}_0 = \mathbb{N}_0 \cup \{\infty\}$ and distribution given by $$ \mathbb{P}(\hat{\xi} = k) = \begin{cases} k \mathbb{P}(\xi = k), & k < \infty \\ 1 - \mathbb{E}[\xi], & k = \infty.\end{cases} $$ Here the topology on $\bar{\mathbb{N}}_0$ is given by the one-point compactification of $\mathbb{N}_0$. Let $d \ge 2$ and $0 \le a < d-1$ be arbitrary integers. My question is if it holds that $$ \lim_{n \to \infty} \mathbb{P}(d^+_{\mathcal{T}_n}(o) \equiv a \mod d) = \mathbb{P}(\hat{\xi} \in a + d\mathbb{Z}) / \mathbb{E}[\xi]. $$ Or if this holds under some additional conditions on the offspring distribution, for example requiring the following limit to exist: $$ \lim_{k \to \infty} \mathbb{P}(\xi = k) / \mathbb{P}(\xi = k+1) $$
A related known result (see Chapter 20 in Janson's survey "Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation") is that when $\Omega_n \to \infty$ is a deterministic sequence that tends to infinity sufficiently slowly, then $$ \lim_{n \to \infty} \mathbb{P}(d^+_{\mathcal{T}_n}(o) \equiv a \mod d, d^+_{\mathcal{T}_n}(o)< \Omega_n) = \mathbb{P}(\hat{\xi} \in a + d\mathbb{Z}). $$ But what about $$\mathbb{P}(d^+_{\mathcal{T}_n}(o) \equiv a \mod d, d^+_{\mathcal{T}_n}(o) \ge \Omega_n)?$$
