Permutation Martingales Overview

• Permutation martingales are sequences derived from random permutations or sampling without replacement that enable systematic analysis of combinatorial statistics.
• They employ explicit conditional expectations and recursive relations to establish central limit theorems and Berry–Esseen bounds for permutation statistics.
• Applications include deriving maximal and rearrangement inequalities, extending to multivariate statistics, and informing sharp concentration results.

A permutation martingale is a martingale sequence naturally associated with random permutations or sampling without replacement schemes. These martingale constructions provide powerful probabilistic tools for analyzing combinatorial statistics of permutations, quantifying deviations and establishing limit theorems. The theory connects strong maximal inequalities for permuted sums to central limit theorems (CLTs) for permutation statistics, unifying classical combinatorial probability with modern martingale methods (Özdemir, 2019, Pozdnyakov et al., 2012).

1. Foundational Construction: Filtrations and Conditional Expectations

Permutation martingales arise from randomizing over all permutations $\pi$ of a finite set (or equivalently, sampling without replacement from a multiset $X = \{x_1, \ldots, x_n\}$) and considering the filtration determined by the sequence of observed elements. Formally, let $\mathcal{F}_k = \sigma(X_1, \ldots, X_k)$ be the filtration, where $X_k = x_{\sigma(k)}$, with $\sigma$ chosen uniformly at random.

Key statistics include:

• $S_k = X_1 + \cdots + X_k$ (partial sums)
• $T_k = X_1^2 + \cdots + X_k^2$ (partial second moments)
• $U_k = X_1^3 + \cdots + X_k^3$, etc.

Conditional expectations are computed explicitly using the uniformity of the draw:

• $$\mathbb{E}[X_{k+1}|\mathcal{F}_k] = \frac{M - S_k}{n - k}$$, where $M = S_n$.
• $$\mathbb{E}[X_{k+1}^2|\mathcal{F}_k] = \frac{B - T_k}{n - k}$$, with $B = T_n$.

This framework allows construction of martingales through systematic linear algebraic recipes: by embedding recursion relations for statistics (possibly vector-valued) and telescoping their evolution (Pozdnyakov et al., 2012).

2. Systematic Martingale Construction and Examples

A general recipe is given for martingale construction:

• Choose a vector of statistics $R_k = (R_k^{(1)}, \ldots, R_k^{(d)})^\top$, all $\mathcal{F}_k$-measurable.
• Identify a deterministic matrix recurrence: $\mathbb{E}[R_{k+1}|\mathcal{F}_k] = A_{k+1} R_k$ (plus possibly an inhomogeneous term).
• Invert $A_{k+1}$ recursively to produce $M_k = A_1^{-1} \cdots A_k^{-1} R_k$, ensuring $\{M_k\}$ is a (vector) martingale.
• Extract coordinates or (adapted) linear combinations to obtain scalar martingales.

Notable explicit constructions:

• Quadratic and bi-quadratic martingales: Take $R_k = (S_k^2, S_k, T_k, 1)^\top$. The “bi-quadratic” martingale,

$$M_k = \frac{(n-k)(n-k-1)}{n(n-1)}\left[(n-1) S_k^2 - k(B - T_k)\right], \quad 1 \leq k \leq n-2,$$

is central when the total sum $M$ vanishes.

• Weighted sum martingales: For fixed weights $a_1, \ldots, a_n$, let $W_k = \sum_{i=1}^k a_i X_i$, and

$M_k = W_k + \frac{a_1 + \cdots + a_k}{n-k} S_k,$

a scalar martingale for $1 \leq k < n$.

These constructions systematically yield martingale difference sequences with explicit increments and controlled variances (Pozdnyakov et al., 2012).

3. Central Limit Theorems for Permutation Statistics via Martingales

Permutation martingales underlie a unified approach to normal approximations for a large family of permutation statistics. For statistics constructed by insertion recurrences—such as descents, inversions, cycles, and statistics for signed or two-sided permutations—one builds a suitable martingale $M_n$ whose increments $d_i$ have variance and moment properties that support CLTs.

The Berry–Esseen bound is extended to martingales with time-dependent (polynomially growing) variances. For $\{d_i\}$ with conditional variance $\sigma_i^2$ and $s_n^2 = \sum_{i=1}^n \sigma_i^2$, if fourth moment and variance growth conditions are satisfied, the following uniform error bound holds:

$$\sup_x |\mathbb{P}[(M_n - \mathbb{E} M_n)/s_n \leq x] - \Phi(x)| \leq C n^{-1/2}$$

for some universal constant $C$ (Özdemir, 2019).

Notable example: for descents $D_n(\pi) = \#\{i : \pi(i) > \pi(i+1)\}$,

• $\mathbb{E}[D_n] = (n-1)/2$, $\operatorname{Var}(D_n) = (n+1)/12$,
• The appropriately normalized descent statistic converges to normality with optimal $O(n^{-1/2})$ Kolmogorov distance.

The same techniques generalize to signed permutations, Stirling permutations, matchings, alternating runs, and joint statistics (e.g., two-sided Eulerian), always yielding near-universal Berry–Esseen rates (Özdemir, 2019).

4. Maximal Inequalities for Permuted Sums

Permutation martingales have enabled new and sharp maximal $L^2$ inequalities for sums and weighted sums of permuted data, generalizing and sometimes improving classical rearrangement inequalities such as those due to Garsia.

Key results include:

• Max-Averages Inequality: For centered data ($\sum x_i = 0$),

$$\frac{1}{n!} \sum_\sigma \max_{1 \leq k \leq n} \left|\frac{1}{k} \sum_{i=1}^k x_{\sigma(i)}\right|^2 \leq \frac{4}{n} \sum_{i=1}^n x_i^2.$$

• Garsia’s Maximal Inequality (Martingale proof):

$$\frac{1}{n!} \sum_\sigma \max_{1 < k < n} \left| \sum_{i=1}^k x_{\sigma(i)} \right|^2 \leq 10 \cdot \frac{1}{n-1} \sum_{i=1}^n x_i^2$$

(improved to constant 2 by later combinatorial methods).

• Quadratic permutation inequalities using bi-quadratic martingales:

$$\frac{1}{n!}\sum_\sigma \max_{2 \leq k \leq n} \left| \frac{(n-1) S_k - (n-k) T_k}{k(k-1)} \right|^2 \leq 4 \cdot \frac{B^2 - Q}{n(n-1)}$$

with $Q = \sum x_i^4$ (Pozdnyakov et al., 2012).

Variation of weights and combinatorics (e.g., alternating sign weights, symmetric bridges) yield sharp constants depending on the cancellation structure, sometimes beating classical inequalities by orders of magnitude.

5. Applications and Extensions

Martingale methods in the permutation context have broad utility in:

• Rearrangements of orthogonal series: Permutation inequalities provide pointwise control over rearranged orthonormal expansions, critical in convergence theory.
• Combinatorial probability and urn models: Tight deviation bounds for hypergeometric-type random variables and urn-based models follow via martingale concentration.
• Discrete bridges and random walks: Uniform control of the deviation from expected parabolic profiles in discrete bridges is a direct application (e.g., in enumeration of Brownian bridges).
• Matrix and vector-valued extensions: Permuting vector or matrix-valued entries and applying the same systematic martingale constructions yield new matrix concentration inequalities (Pozdnyakov et al., 2012).

A plausible implication is that these martingale techniques may be adaptable for sharp weak-type ($L^1$ or weak-$(1,1)$) maximal inequalities, and for higher-order moments by enlarging the vector of tracked statistics.

6. Comparison with Classical and Modern Inequalities

Permutation martingales streamline, generalize, and sometimes quantitatively improve classical results:

• Garsia’s rearrangement inequalities (constant 80 in the weighted sum setting) are rederived via martingale algebra in a few lines, with explicit constants.
• Chobanyan & Salehi’s combinatorial refinements push constants to optimal values using martingales and exchangeability arguments, matching the structure of permutation martingale methods but with optimal calibration.
• New results include a Hardy-type average bound with constant $4/n$ and previously unrecorded bi-quadratic permutation inequalities.

Table: Comparison of Key Inequalities

Inequality Type	Martingale Constant	Best Known Constant	Reference
Unweighted Garsia (max partial sum)	10	2	(Pozdnyakov et al., 2012), Chobanyan/Salehi
Hardy max-average	4/n	4/n (new)	(Pozdnyakov et al., 2012)
Weighted sum (Garsia)	80	2	(Pozdnyakov et al., 2012), Chobanyan/Salehi

This suggests that systematic martingale constructions not only subsume earlier probabilistic and combinatorial approaches but can be tuned, by the choice of tracked statistics and weights, to yield broader and sharper results than previously possible.

7. Overview of Further Developments

Permutation martingales form a versatile platform for quantitative combinatorial probability, underpinned by explicit couplings and recursions. Extant research has developed:

• High-dimensional and multivariate CLTs for joint permutation statistics (e.g., descents and inverse descents)
• Extensions to power sums of higher order, cross terms, and noncommutative cases using symbolic algebra for matrix recursions.
• Applications to random matrix theory via generalizations to hypergeometric matrix ensembles.

The methods and results of Pozdnyakov & Steele (Pozdnyakov et al., 2012) and Fulman et al. (Özdemir, 2019) have become foundational references for future explorations in combinatorial martingale methods, permutation limit theorems, and maximal inequalities for dependent structures.