Table with the most seated customers in Chinese restaurant process

Question

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the probability that my favourite table would be the one with most people after, say, $10$ rounds? I am aware that this might be somewhat complex to calculate so I would also be interested to hear about some good estimates or relevant work.

For those who are not familiar with Chinese restaurant process, it goes as follows. Let $\alpha>0$ be some parameter. At each round, a new customer enters the restaurant. He is seated to a new table with probability $\frac{\alpha}{\alpha+n}$ and at table $k$ with probability $\frac{n_k}{\alpha+n}$ where $n_k$ is the number of customers already at table $k$, and $n$ is the total number of customers already seated (hence, $n=\sum_k n_k$).

Answer 1

To restate your problem in technical terms: What is the approximation of the distribution of order statistics (in OP we can focus on maximum statistics only) of sample from a Dirichlet process of size $n=10$? In other words, we need to check the distribution of $(X_{(1)},\cdots X_{(10)})$ We approach this problem in two aspects.

The first is to consider a small sample approximation to the distribution of $(X_{(1)},\cdots X_{(10)})$. This is fully discussed in Sec 4 of [Stigler].

The second is to consider an even extreme case, what if we let the sample size $n\rightarrow \infty$ and check the distribution of the order statistics. The core concept in solving this problem in [Stigler] is the so-called order-statistic process. Given a Dirichlet process $\boldsymbol{F}(t)$ on $[0,1]$ indexed by a base measure $\nu([0,t])=\lambda t$ we call this Dirichlet process a uniform order-statistics process because it describes the distribution of uniform distribution.

For a finite sample $n=10$ from (another) Dirichlet process we can actually derive their joint distribution $G$ and then the induced order-statistics process is $G^{-1}(\boldsymbol{F}(t))$. The main result is the Gaussian first order approximation.

[Stigler] Theorem 1. If $\alpha(t)=\frac{1}{\lambda}\nu(t)$ is continuous on $[0,1],\lambda=\nu([0,1])$ then $\boldsymbol{Z}_{\lambda}(t)=\sqrt{1+\lambda}\left(\boldsymbol{F}(t)-\alpha(t)\right)$ converges in distribution as $\lambda\rightarrow\infty,\alpha(t) $ fixed, to the Gaussian process $\boldsymbol{Z}(t)$ on $[0,1]$ with zero mean and $cov[\boldsymbol{Z}(s),\boldsymbol{Z}(t)]=\alpha(s)[1-\alpha(t)]$.

In other words, if we choose $\nu$ as counting measure with sample size $n$, then this theorem literally describe the asymptotic behavior of the order statistics, hence the maximum as $n\rightarrow\infty$.

While an earlier work [Fan&Liu] achieved first-order accuracy for general processes, a recent work by [Goldman&Kaplan] answered this question in higher order accuracy for i.i.d observations(not the case in OP). So this fractional order statistics theory actually shed some light in quantile inference in recent research.

Reference

[Fan&Liu]Fan, Yanqin, and Ruixuan Liu. "A direct approach to inference in nonparametric and semiparametric quantile models." Journal of Econometrics 191.1 (2016): 196-216.

[Goldman&Kaplan]Goldman, Matt, and David M. Kaplan. "Fractional order statistic approximation for nonparametric conditional quantile inference." Journal of Econometrics 196.2 (2017): 331-346.

[Stigler]Stigler, Stephen M. "Fractional order statistics, with applications." Journal of the American Statistical Association 72.359 (1977): 544-550.