Let $F_0$ be some probability measure and $\alpha > 0$ be the concentration parameter. I can draw a random distribution from $F\sim \mathrm{DP}(\alpha, F_0)$ using the stick-breaking construction: \begin{align*} v_k &\sim \mathrm{Beta}(1, \alpha)\\ \phi_k &\sim F_0\\ \pi_k &= v_k \prod_{j < k} (1-v_j)\\ F &= \sum_{k=1}^{\infty} \pi_k \delta_{\phi_k}. \end{align*}
Now consider a finite-dimensional Dirichlet distribution with $K$ classes. If I draw a random distribution $F^K$ according to the following model: \begin{align*} \pi' &\sim \mathrm{Dirichlet}(\alpha/K, \cdots, \alpha/K)\\ \phi'_k &\sim F_0\\ F^K &\sim \sum_{k=1}^K \pi'_k \delta_{\phi'_k}, \end{align*} then for every function $f$ integrable with respect to $F_0$ it holds that the sequence of random variables $\mathbb E_{F^K}[f]$ converge in distribution to $\mathbb E_F[f]$ as $K\to \infty$.
Hence, I'd naively expect that several entries of $\pi'$ for very large $K$ should be approximately the same as several entries of $\pi$.
However, I have a trouble to understand why it mathematically would be the case. Recall that sampling $\pi'\sim \mathrm{Dirichlet}(\alpha/K, \cdots, \alpha/K)$ can be accomplished by a stick-breaking construction (valid for $k = 1, \dotsc, K-1$): \begin{align*} u_k & \sim \mathrm{Beta}(\alpha/K, \alpha\cdot (1-k/K) )\\ \pi'_k &= u_k \prod_{j<k} (1-u_j) \end{align*}
For $k \ll K$ I have $1-k/K \approx 1$ and this stick-breaking construction looks deceptively similar, apart from the fact that I sample from $$u_k \sim \mathrm{Beta}(\alpha/K, \alpha)$$ rather than $$v_k \sim \mathrm{Beta}(1, \alpha)$$ used in the stick-breaking process of the Dirichlet process.
I have been thinking quite a lot about this discrepancy and I realized that as I have a symmetric distribution $\mathrm{Dirichlet}(\alpha/K, \cdots, \alpha/K)$, there is no reason why e.g., $\pi'_1$ should match as $\pi_1$ (rather than, say, $\pi'_5$). Hence, the stick-breaking process of $\pi'$ above doesn't really mimic well the strick-breaking process used to sample $\pi$.
I have tried to read J. Pitman's "Random Discrete Distributions Invariant under Size-Biased Permutation", but I have not really understood how to improve the stick-breaking process of $u_k$ to make it size-biased.
Hence, my question is:
Is there a size-biased stick-breaking construction of the symmetric Dirichlet distribution which would mimic the one of Dirichlet process?
References studying this construction would be very welcome!
