Discrete Mass-Distribution Function

• Discrete mass-distribution functions are defined as functions that assign weights to countable points, resulting in purely atomic measures with stepwise cumulative distributions.
• They enable precise integration by summing over atomic contributions and support robust estimation techniques such as maximum likelihood and spectral reconstruction.
• They have broad applications in statistical mechanics, swarm robotics, and network analysis, making them essential for modeling discrete probabilistic events.

A discrete mass-distribution function assigns probability or weight to a countable set of points in the underlying space, giving rise to stepwise constant cumulative functions and purely atomic measures. This concept is foundational in probability theory, statistical modeling, integration theory, statistical mechanics of fragmentation, swarm robotics, and multivariate asymptotic analysis. Discrete mass-distribution functions enable precise encoding of probabilistic laws for inherently discrete outcomes and serve as building blocks for modeling and computation in systems ranging from stochastic processes to high-dimensional network flows.

1. Formal Definition and Mathematical Structure

A discrete mass-distribution function is typically formalized in terms of a probability mass function (pmf) or a pure-jump cumulative distribution function (cdf). Given a countable set of atoms $\{x_i\}_{i\in I}$, a function $F: \mathbb{R} \to \mathbb{R}$ is a discrete mass-distribution function if:

• $F$ is non-decreasing and right-continuous;
• $F$ is constant on each open interval between its atoms;
• Each $x_i$ is a jump point at which $\Delta F(x_i) := F(x_i) - F(x_i^-)>0$.

The measure $\mu_F$ generated by $F$ is purely atomic, assigning mass $\Delta F(x_i)$ to $x_i$ and zero elsewhere. The associated pmf is $p(x_i) = \Delta F(x_i)$, and $\sum_{i\in I} p(x_i) = 1$ if $F$ is the cdf of a probability law (Niang et al., 2020).

In multivariate settings, a discrete mass function on $\mathbb{N}^d$ is a function $f:\mathbb{N}^d\to [0,\infty)$ with $\sum_{k\in\mathbb{N}^d} f(k)=1$. Such an $f$ induces the purely atomic measure $\mu$: $\mu(\{k\})=f(k)$ (Wang et al., 2016).

2. Integration and Measure-Theoretic Foundations

Integration with respect to discrete mass-distribution functions realizes Riemann–Stieltjes and Lebesgue–Stieltjes integrals as atomic sums. For $f$ bounded and continuous at each atom $x_i$, the Riemann–Stieltjes integral simplifies to

$$\int_{a}^{b} f(x)\,dF(x) = \sum_{x_i \in (a,b]} f(x_i) \Delta F(x_i),$$

matching the Lebesgue–Stieltjes atomic sum

$$\int_{a}^{b} f(x)\,d\mu_F(x) = \sum_{x_i \in [a,b]} f(x_i) \Delta F(x_i)$$

whenever $\sum_{i} |f(x_i)|\Delta F(x_i)<\infty$ (Niang et al., 2020).

This universality underlies the role of discrete mass-distribution functions in probability and information theory, as their atoms fully encode the statistical law; all expectations, moments, and higher-order statistics are summable over this discrete support.

3. Canonical Examples and Parametric Families

A variety of probabilistically meaningful and highly flexible mass-distribution functions exist in the literature:

• Discrete Two-Sided Power (DTSP) Distribution: For $Y\sim \mathrm{DTSP}(a,m,b,n)$, the pmf on $\mathcal{S} = \{a,\dots,b-1\}$ is defined in two pieces (see formula block in (Chakraborty et al., 2015)). The shape parameter $n>0$ interpolates among uniform, triangular, trapezoidal, $J$-, $U$-, strictly increasing/decreasing, and bathtub forms. Survival and hazard functions admit closed-form expressions:

$$S(y) = \frac{(m-a)^{n}-(y-a)^{n}}{(b-a)(m-a)^{n-1}} \text{ for } y\le m,\quad h(y) = \frac{p(y)}{S(y)}.$$

The DTSP nests multiple classic mass-distribution families and is especially effective for modeling count data with non-monotonic hazard rates (Chakraborty et al., 2015).

• Discrete Gauss–Poisson Law: The pmf

$$p_{GP}(n;z,r) = [R(r;z)]^{-1} \frac{e^{zn}}{n!} e^{-\frac{1}{2}r n^2}$$

with normalization $$R(r;z) = \sum_{n=0}^\infty \frac{e^{zn}}{n!} e^{-\frac{1}{2}r n^2}$$, interpolates between Poisson ($r=0$) and degenerate ($r\to\infty$) laws. It arises naturally in statistical physics, notably in Curie–Weiss cell models with both mean-field attraction and on-site repulsion (Dobush et al., 2024).

• Multicomponent Fragmentation: The probability of a specific fragmented configuration, $p(\{n_i^\alpha\})$, is proportional to its multiplicity $\Omega(\{n_i^\alpha\})$, where combinatorial structures can be computed in closed form. Introducing bias functionals enables systematic tuning of fragment mixing or segregation (Matsoukas, 2020).

4. Estimation, Computational Techniques, and Representations

Estimation for discrete mass-distribution functions is typically performed via:

• Maximum Likelihood Estimation (MLE): For parametric mass functions, log-likelihoods in the parameters (e.g., shape parameter $n$ in DTSP) are optimized numerically, often via root-finding or Newton–Raphson (Chakraborty et al., 2015).
• Method of Moments: Model moments are equated with sample moments. For the DTSP, the raw $k$th moment is given by

$E[Y^k] = \sum_{y=a}^{b-1} y^k p(y),$

with closed-form expressions in terms of generalized harmonic numbers when possible (Chakraborty et al., 2015).

• Spectral Reconstruction: For lattice-supported distributions, probability masses $p_n$ can be rapidly recovered from the characteristic function via the inverse discrete Fourier transform (iDFT) and fast Fourier transform (FFT), with explicit control over approximation errors, leveraging resultants from either moment bounds or eventual monotonicity (Warr, 2012).
• Decentralized Mass Estimation: In swarm robotics, each agent employs local kernel-based contributions and inter-agent consensus dynamics to estimate the mass at each sample point, ensuring convergence to the global mass vector of the full system (Cai et al., 1 Feb 2026).

5. Applications Across Domains

• Statistical Physics and Fragmentation: Statistical mechanics of fragmentation events for multicomponent systems is phrased entirely in the language of discrete mass-distribution functions. The distribution over fragment compositions follows from multiplicity-weighted ensembles, and generalizations allow for arbitrary biasing of fragmentation pathways (Matsoukas, 2020). In the Curie–Weiss cell model, the discrete Gauss–Poisson distribution quantifies particle occupancy statistics and relates directly to phase transitions in the system's thermodynamics (Dobush et al., 2024).
• Swarm Configuration and Formation Control: The discrete mass-distribution function defines a target pattern by associating desired density (mass) to a collection of sample points. Distributed meanshift control laws use this structure to enable robust and adaptive shape formation, especially useful for decentralized multi-robot systems with variable population size or complex target geometries (Cai et al., 1 Feb 2026).
• Stochastic Processes, Integration, and Data Analysis: Discrete mass-distribution functions serve as the basis for integrating observables, computing expectations, and constructing stochastic models where outcomes are intrinsically discrete (e.g., event counts, arrival times, lattice-based processes) (Niang et al., 2020, Warr, 2012).
• Network Analysis and Multivariate Regular Variation: The embedding methodology for regular variation ensures that discrete mass functions for network in-/out-degree distributions inherit multivariate power-law structure, thereby allowing discrete observables to be analyzed with foundational continuous probabilistic machinery (Wang et al., 2016).

6. Structural Properties and Flexibility

Key properties of discrete mass-distribution functions include:

Property	Example/Reference	Significance
Arbitrary support shapes	DTSP, fragmentation	Supports complex, non-uniform, and multi-modal distribution shapes
Closed-form moments	DTSP, Gauss–Poisson	Enables precise summary statistics
Rich parameterization	DTSP: $n$, $a$, $m$, $b$; Gauss–Poisson: $z$, $r$	Flexible modeling of behaviors from uniformity to extreme localization
Decentralized adaptability	Swarm formation	Robust to population changes and suited for decentralized computation

Parametric families such as the DTSP allow modeling of failure rates and event probabilities for data exhibiting diverse empirical shapes (bathtub, J, U, etc.), while kernel-based discrete mass distributions facilitate geometric fidelity for arbitrary target sets in distributed systems (Chakraborty et al., 2015, Cai et al., 1 Feb 2026).

7. Asymptotic, Combinatorial, and Multivariate Analysis

• Asymptotics: The discrete Gauss–Poisson normalization $R(r;z)$ displays distinct scaling as $z\to\infty$ with subleading oscillatory corrections for large $r$, a feature traced to the structure of the underlying sum and with parallels to Laplace asymptotics for continuous integrals (Dobush et al., 2024).
• Combinatorial Enumeration: In fragmentation, closed-form partition functions and multiplicities for mass-distribution functions are obtained via multinomial and hypergeometric identities, providing both exact means and explicit forms for modified (biased) fragmentation scenarios (Matsoukas, 2020).
• Regular Variation and Embeddability: Discrete mass functions on $\mathbb{N}^d$ can be embedded in regularly varying continuous densities provided monotonicity or convergence-on-the-sphere conditions hold. This allows for inherited scaling limits and tail behavior essential for the analysis of large networks and heavy-tailed phenomena (Wang et al., 2016).

A plausible implication is that the discrete mass-distribution function concept unifies methods ranging from applied combinatorics to decentralized multi-agent control, centering on the specification and manipulation of atomic probability masses for discrete events or configurations. Its theoretical tractability, computational accessibility, and flexibility make it indispensable to mathematical statistics, physics, and systems engineering.