Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

Question

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research literature is "because it produces power laws", pointing out a connection to Zipf's law which says that ordered word frequencies follow a power law.

From my current understanding (please correct me, if I'm wrong), this translates into a claim about the tails of the Dirichlet-Poisson distribution:

Claim 1: Let $\boldsymbol{\tilde{\pi}}\sim\mbox{PD}\left(\alpha,\theta\right)$ with $\alpha\in(0,1)$ and $\theta>-\alpha$ Then

$$\tilde{\pi}_{n}/n^{-\lambda}\to X\mbox{ a.s.}$$

for some bounded random variable $X$ and $\lambda > 0$.

My current understanding:

• My starting point is the usual stick-breaking construction of the Pitman-Yor process with discount parameter $\alpha\in(0,1)$, concentration parameter $\theta>-\alpha$, and non-atomic base measure $H\in\mathcal{M}_{1}\left(V\right)$:

  $$G:=\sum_{n\in\mathbb{N}}\pi_{n}\delta_{\phi_{n}}\mbox{ with }\boldsymbol{\pi}\sim\mbox{GEM}\left(\alpha,\theta\right)\mbox{ and }\phi_{n}\overset{\operatorname{iid}}{\sim}H,$$ where $\boldsymbol{\pi}$ can be represented as a stick breaking sequence: $\pi_{n}=V_{n}\prod_{k=1}^{n-1}\left(1-V_{k}\right)$ with $V_{k}\sim\mbox{Beta}\left(1-\alpha,\theta+k\alpha\right)$.

• Drawing from G, $\theta_{1},\ldots,\theta_{n}\overset{\operatorname{iid}}{\sim} G$, gives rise to a sequence of exchangeable partitions $\left(\Pi_{n}\right)_{n\in\mathbb{N}}$ given by the equivalence relation $i\sim j\iff\theta_{i}=\theta_{j}$. The empirical distribution of partition sizes $$\nu_{n}=\sum_{k=1}^{\infty}P_{k}^{\left(n\right)}\delta_{k}$$ with $P_{k}^{\left(n\right)}=n^{-1}\left|\left\{ A\in\Pi_{n}:\left|A\right|=k\right\} \right|$ almost surely converges against $\mbox{GEM}\left(\alpha,\theta\right)$.

• The distribution of the ordered sequence $\tilde{\pi}_{1}\geq\tilde{\pi}_{2}\geq\ldots$ of stick-breaking weights $\pi_{1},\pi_{2},\ldots$ is called $\mbox{PD}\left(\alpha,\theta\right)$.

Questions:

1. Is my understanding correct, that is: Is the PY process chosen in language modeling because claim 1 holds? If yes, could you please point me to a proof? (Potential hit: Lemma 3.11. of Pitman, Combinatorial Stochastic Processes - if the PY process gives rise to an $\left(\alpha,\theta\right)$-partition. He doesn't prove the lemma and I'd expect that a property of the PY process, which is taken for granted in so many papers, has an accessible proof.)

2. To which extent does this reasoning work if I pick a base measure $H$ that has point masses? Will the empirical distribution of partition sizes $\nu_{n}=\sum_{k=1}^{\infty}P_{k}^{\left(n\right)}\delta_{k}$ still converge against $\mbox{GEM}\left(\alpha,\theta\right)$?

Answer 1

As an intermediate solution, I put up with the following proposition. It only gives the asymptotic behavior of the expectation of the stick-breaking weights while almost surely results exist, for sure.

Proposition: Let $\left(\pi_{k}\right)_{k\in\mathbf{N}}$ be the stick-breaking weights of a Pitman-Yor process with concentration parameter $\alpha$ and discount parameter $d$, that is $\boldsymbol\pi\sim\mbox{GEM}(\alpha,d)$. Then

$$\mathbf{E}\left[\pi_{k}\right]\in\begin{cases} O\left(k^{-1/d}\right) & \mbox{for }d>0\\ O\left(\left(\frac{\alpha}{\alpha+1}\right)^{k}\right) & \mbox{for }d=0. \end{cases} $$

Statement in plain English: In the case of the Dirichlet process, $\boldsymbol{\pi}\sim\mbox{GEM}\left(\alpha,0\right)$, the stick-breaking weights $\pi_{k}$ decrease on average exponentially with $k$. In the case of the Pitman-Yor process, $d>0$, the tails of the sequence $\boldsymbol{\pi}\sim\mbox{GEM}\left(\alpha,d\right)$ are on average those of a power law with exponent $-1/d$. The higher the discount parameter, the heavier the tails.

Proof: Recall that $\pi_{k}=V_{k}\prod_{n=1}^{k-1}\left(1-V_{n}\right)$ for independent $V_{n}\sim\mbox{Beta}\left(1-d,\alpha+nd\right)$. By independence and linearity of the expectation we have

$$\mathbf{E}\left[\pi_{k}\right]=\frac{1-d}{1+\alpha+\left(k-1\right)d}\prod\frac{\alpha+nd}{1+\alpha+\left(n-1\right)d}. $$

For $d=0$ , this boils down to

$$\mathbf{E}\left[\pi_{k}\right]=\frac{1}{1+\alpha}\left(\frac{\alpha}{1+\alpha}\right)^{k-1}. $$

For $d>0$, we represent the raising factorials $\left(\alpha\right)_{n,d}$ and $\left(1+\alpha\right)_{n,d}$ by Gamma functions,

$$\mathbf{E}\left[\pi_{k}\right] = C\frac{\Gamma\left(\alpha/d+\left(k-1\right)\right)}{\Gamma\left(\left(1+\alpha\right)/d+\left(k-1\right)\right)}.$$

Using the asymptotic ratio property of the Gamma function, $\frac{\Gamma(x+\epsilon)}{\Gamma(x)}\to x^{-\epsilon}$ with $\epsilon=1/d$, yields the claim.