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The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis

Guy Louchard, Werner Schachinger, Mark Daniel Ward

Abstract

The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.

The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis

Abstract

The analysis of strings of random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic () mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.
Paper Structure (38 sections, 18 theorems, 222 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 38 sections, 18 theorems, 222 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Further conjectures and results for pairs of identical letters
  4. Higher moments
  5. Limiting distribution
  6. The probability of avoiding certain pairs via Markov chains.
  7. Two pairs $(i,i)$ and $(r,r)$ of identical letters
  8. The variance of $X^{(n)}_1$
  9. Contribution of covariances to the variance of $X^{(n)}_2$
  10. More than two pairs of identical letters
  11. Some further results on the probability of avoiding a prescribed set of pairs
  12. Limiting distribution of $X^{(n)}_3$
  13. Combinatorial Pattern Matching Approach
  14. Main results
  15. Analysis of the average number of distinct (adjacent) pairs
  16. ...and 23 more sections

Key Result

Theorem 2.1

Let $L:=\ln(1/q)$ and $\chi:=2\pi\mathbf{i}/L$, where $\mathbf{i}$ denotes the imaginary unit. We also define where $F_1(s)=\sum_{i,k\ge1}\left [(q^i+q^k-pq^{i+k-1})^{-s}-(q^i+q^k)^{-s}\right ]$, and the constant term of $T^{(n)}_2$ simplifies to Then, as $n\to\infty$, the variances of $X^{(n)}_i$, $1\le i\le 3$, satisfy

Figures (3)

  • Figure 1: Plot of $2(1-q)F'_1(0)$, showing the dependence of the constant term $\frac{2}{L}F'_1(0)$ on $q$. We leave it as an exercise to show that, for $q\to0$ (resp. $q\to1$), the limit is $2\ln 2$ (resp. $4\ln 2$).
  • Figure 2: Comparison between $f(\eta)$ (line) and the simulation of $X^{(n)}_1$ (circles), $p=1/4$, number of simulated words $=50000$ for each $n\in\{10000,11547,13333,15396\}$.
  • Figure 3: Comparison between Gaussian density $f(x)$ (line) and the simulation of $X^{(n)}_3$ (circles), with $p=1/4,n=500000$, and number of simulated words $N=200000$.

Theorems & Definitions (27)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Conjecture 3.1
  • Theorem 3.2
  • Remark 3.3
  • Conjecture 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 17 more