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Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Abstract

Let \(\xi _1,\xi _2,\ldots \) be a sequence of independent copies of a random vector in \(\mathbb {R}^d\) having an absolutely continuous distribution. Consider a random walk \(S_i:=\xi _1+\cdots +\xi _i\), and let \(C_{n,d}:={{\,\mathrm{conv}\,}}(0,S_1,S_2,\ldots ,S_n)\) be the convex hull of the first \(n+1\) points it has visited. The polytope \(C_{n,d}\) is called k-neighborly if for any indices \(0\le i_1<\cdots < i_k\le n\) the convex hull of the k points \(S_{i_1},\ldots , S_{i_k}\) is a \((k-1)\)-dimensional face of \(C_{n,d}\). We study the probability that \(C_{n,d}\) is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to \(\infty \). There is an explicit formula for the expected number of \((k-1)\)-dimensional faces of \(C_{n,d}\) which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of \(C_{n,d}\). This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.

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Notes

  1. Note that the claim of Theorem VIII.8 holds locally uniformly in \(\lambda \).

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Acknowledgements

ZK acknowledges support by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics-Geometry-Structure and by the DFG priority program SPP 2265 Random Geometric Systems. AM was supported by the National Research Foundation of Ukraine (Project 2020.02/0014 “Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability”). The authors thank Thomas Godland for useful discussions and the anonymous referee for useful suggestions.

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Appendix

Appendix

1.1 Proof of Theorem 5.3

Recall that the distribution of the random uniform composition \((b_1^{(n)},\ldots ,b_k^{(n)})\) can be represented as

$$\begin{aligned} \mathbb {P}[(b_1^{(n)},\ldots ,b_k^{(n)})\in \cdot ]=\mathbb {P}[(G_1,\ldots ,G_k)\in \cdot |G_1+\cdots +G_k=n], \end{aligned}$$

where \(G_1,\ldots ,G_k\) are independent random variables having the same geometric law on \(\mathbb {N}\) with parameter \(\theta \). This representation holds for arbitrary \(\theta \in (0,1)\) and we are free to choose \(\theta :=\theta _n=k/n\). As we demonstrated in Lemma 5.5, it suffices to show that

$$\begin{aligned} \left( \frac{N_j^{(n)}-k\theta _n(1-\theta _n)^{j-1}}{\sqrt{k}}\right) _{j\ge 1}\overset{d}{\underset{n\rightarrow \infty }{\longrightarrow }}\left( \mathscr {N}_j\right) _{j\ge 1}. \end{aligned}$$

By the Cramér–Wold device the last display is equivalent to

$$\begin{aligned} \frac{\sum _{l=1}^{M}\beta _l (N_l^{(n)}-k\theta _n(1-\theta _n)^{l-1})}{\sqrt{k}}\overset{d}{\underset{n\rightarrow \infty }{\longrightarrow }}\sum _{l=1}^{M}\beta _l \mathscr {N}_l, \end{aligned}$$

for arbitrary fixed \(M\in \mathbb {N}\) and \(\beta _1,\beta _2,\ldots ,\beta _M\in \mathbb {R}\). Put

$$\begin{aligned} f_n(x):=\sum _{l=1}^{M}\beta _l (\mathbbm {1}_{\{x=l\}}-\theta _n(1-\theta _n)^{l-1}),\quad x\in \mathbb {N}, \end{aligned}$$

and, further,

$$\begin{aligned} S_{n,k}:=\sum _{j=1}^{k}G_j,\quad T_{n,k}:=\sum _{j=1}^{k}f_n(G_j). \end{aligned}$$

The subsequent analysis relies on the following representation:

$$\begin{aligned}&{{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t k^{-1/2} \left( \sum _{l=1}^{M}\beta _l (N_l^{(n)}-k\theta _n(1-\theta _n)^{l-1})\right) \right) ={{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t k^{-1/2} \sum _{j=1}^k f_n(b_j^{(n)})\right) \\&\quad \overset{(5.2)}{=}{{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t k^{-1/2} \sum _{j=1}^k f_n(G_j)\Big |S_{n,k}=n\right) ={{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t k^{-1/2} T_{n,k}\Big |S_{n,k}=n\right) . \end{aligned}$$

Thus, it suffices to prove that, for every fixed \(t\in \mathbb {R}\),

$$\begin{aligned} \lim _{n\rightarrow \infty }{{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t k^{-1/2} T_{n,k}\Big |S_{n,k}=n\right) ={{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}t \sum _{l=1}^{M}\beta _l \mathscr {N}_l\right) . \end{aligned}$$
(8.1)

According to Theorem 1 in [33] we have

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left( \mathrm{e}^{\mathrm{{i}}t k^{-1/2}T_{n,k}}\Big | S_{n,k}=n\right)&= \frac{1}{2\pi \mathbb {P}[S_{n,k}=n]}\int _{-\pi }^{\pi }{{\,\mathrm{\mathbb {E}}\,}}\mathrm{e}^{\mathrm{{i}}s (S_{n,k}-n)+\mathrm{{i}}tk^{-1/2}T_{n,k}}\mathrm{d}s\nonumber \\&=\frac{1}{2\pi \sqrt{k}\mathbb {P}[S_{n,k}=n]}\int _{-\pi \sqrt{k}}^{\pi \sqrt{k}}{{\,\mathrm{\mathbb {E}}\,}}\mathrm{e}^{\mathrm{{i}}u k^{-1/2}(S_{n,k}-n)+\mathrm{{i}}tk^{-1/2}T_{n,k}}\mathrm{d}u. \end{aligned}$$
(8.2)

Using the Lindeberg–Feller central limit theorem we obtain

$$\begin{aligned} \left( \frac{S_{n,k}-n}{\sqrt{k}},\frac{T_{n,k}}{\sqrt{k}}\right) \overset{d}{\underset{n\rightarrow \infty }{\longrightarrow }}(\widetilde{N}_1,\widetilde{N}_2), \end{aligned}$$

where \((\widetilde{N}_1,\widetilde{N}_2)\) is a centred Gaussian vector with the following variances and covariance:

$$\begin{aligned} \sigma _1^2&:=\mathop {\mathrm {Var}}\nolimits \widetilde{N}_1= \lim _{n\rightarrow \infty }\mathop {\mathrm {Var}}\nolimits (G_1)=\frac{1-\alpha }{\alpha ^2},\\ \sigma _2^2&:=\mathop {\mathrm {Var}}\nolimits \widetilde{N}_2=\lim _{n\rightarrow \infty }\mathop {\mathrm {Var}}\nolimits (f_n(G_1))=\sum _{l=1}^{M}\beta ^2_l\alpha (1-\alpha )^{l-1}-\left( \sum _{l=1}^{M}\beta _l \alpha (1-\alpha )^{l-1}\right) ^2\!, \end{aligned}$$

and

$$\begin{aligned} r&:={{\,\mathrm{Cov}\,}}(\widetilde{N}_1,\widetilde{N}_2)=\lim _{n\rightarrow \infty }{{\,\mathrm{Cov}\,}}(f_n(G_1),G_1))\\&=\lim _{n\rightarrow \infty }{{\,\mathrm{Cov}\,}}\left( \sum _{l=1}^{M}\beta _l \mathbbm {1}_{\{G_1=l\}},\sum _{l=1}^{M}l \mathbbm {1}_{\{G_1=l\}}\right) \\&=\sum _{l=1}^{M}\beta _l l \alpha (1-\alpha )^{l-1}-\alpha ^{-1}\sum _{l=1}^{M}\beta _l \alpha (1-\alpha )^{l-1}=\sum _{l=1}^{M}\beta _l (l\alpha -1)(1-\alpha )^{l-1}. \end{aligned}$$

Since \(S_k-k\) has the negative binomial distribution, direct calculation shows that the limit \(\lim _{n\rightarrow \infty }\sqrt{k}\mathbb {P}[S_k=n]\) exists and is positive. Thus, by the Lebesgue dominated convergence theorem, we deduce from (8.2) that

$$\begin{aligned}&\lim _{n\rightarrow \infty }{{\,\mathrm{\mathbb {E}}\,}}\left( \mathrm{e}^{\mathrm{{i}}t k^{-1/2}T_{n,k}}\Big | S_{n,k}=n\right) =\mathrm{const}\cdot \int _{-\infty }^{\infty }{{\,\mathrm{\mathbb {E}}\,}}\exp \left( \mathrm{{i}}u \widetilde{N}_1+\mathrm{{i}}t\widetilde{N}_2\right) \mathrm{d}{u}\nonumber \\&\quad =\mathrm{const}\cdot \int _{-\infty }^{\infty }\exp \left( -\frac{u^2\sigma _1^2+t^2\sigma _2^2+2rut}{2}\right) \mathrm{d}u=\exp \left( -\frac{\sigma _1^2 \sigma _2^2 -r^2}{2\sigma _1^2}t^2\right) . \end{aligned}$$
(8.3)

To ensure applicability of the dominated convergence (which is non-trivial), one can argue as in the paper of Holst [33] who relies on [52]. It remains to note that the right-hand sides of (8.1) and (8.3) coincide as is readily seen by comparing the variances.

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Kabluchko, Z., Marynych, A. Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks. Probab. Theory Relat. Fields 184, 969–1028 (2022). https://doi.org/10.1007/s00440-022-01146-9

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Keywords

  • Stirling numbers
  • Lah numbers
  • Lah distribution
  • Records
  • Random compositions
  • Random walks
  • Random polytopes
  • Convex hulls
  • Neighborliness
  • f-vectors
  • Mod-Poisson convergence
  • Central limit theorem
  • Large deviations
  • Lambert W-function
  • Threshold phenomena
  • Conic intrinsic volumes
  • Weyl chambers

Mathematics Subject Classification

  • Primary: 11B73
  • 60C05
  • Secondary: 60D05
  • 52A22
  • 52A23
  • 60F05
  • 60F10
  • 30C15
  • 26C10
  • 05A16
  • 05A18