Localized Cumulative Distributions (LCDs)

• Localized cumulative distributions (LCDs) are generalizations of classical CDFs that accumulate probability over localized subsets using kernel weighting and restricted domains.
• They enable sharp statistical bounds and error estimation in tail-risk analysis, contributing to robust frameworks in signal processing, network science, and control theory.
• LCDs support efficient optimization and detection algorithms by providing localized probabilistic assessments essential for non-linear positioning and high-dimensional inference.

Localized cumulative distributions (LCDs) generalize classical cumulative distribution functions by emphasizing probability accumulation over restricted domains, weighted neighborhoods, or localized subsets of data and state spaces. LCDs have become central tools in areas ranging from network science, statistical inference, signal processing, and control theory, driven by the need to quantify and bound probabilities or deviations in spatial, temporal, or high-dimensional settings. The following sections synthesize key developments, mathematical structures, theoretical insights, and modern applications derived from the research literature, with special attention to technical rigor and distinctions arising from domain properties.

1. Fundamental Principles and Mathematical Formulation

LCDs generalize the cumulative distribution function (CDF) by localizing probability with respect to subsets, kernels, or metric neighborhoods. For a random variable $X$ with density $f(x)$, the classical (global) CDF is $F(x) = P(X \le x)$. In LCDs, localization is effected by restricting the domain (e.g., $P(X \le x, X \in \Omega)$), by kernel weighting, or by focusing on specific structures (e.g., neighborhoods in networks, or blocks in ordered score spaces).

A prevalent kernel-weighted LCD for $\mathbf{x} \in \mathbb{R}^d$ is

$F(m, b) = \int_{\mathbb{R}^d} f(x) K(x, m, b)\,dx$

where $K(x, m, b)$ is a positive, often normalized kernel (e.g., Gaussian, $K(x, m, b) = \exp(-\|x - m\|^2 / 2b^2)$), $m$ is the localization center, and $b$ encodes bandwidth. This construction underpins deterministic low-discrepancy sample set generation in multidimensional optimization, as well as localized tail probability analysis (Walker et al., 7 Oct 2025).

In network analysis, LCDs typically sum over local modules:

$$P_{\mathrm{LCD}}(k; \Omega) = \frac{1}{n_v(\Omega)} \sum_{k' > k,\, v \in \Omega} 1$$

where $\Omega$ is the local region or module, and $n_v(\Omega)$ its vertex count (Wang et al., 2016). This facilitates locality-sensitive measurement of scale-freeness and structural heterogeneity.

For center-of-gravity (COG) algorithms in positioning, LCDs originate from cumulative probabilities over complex regions induced by non-linear combinations of random strip responses (Landi et al., 2020, Landi et al., 2021).

2. Domain Structure: Continuous vs. Discrete, Geometrically Growing vs. Uniform Domains

The relationship between the exponent of a power-law distribution and its cumulative counterpart depends critically on the support structure of the random variable. For continuous variables (or discrete ones with unit increments), if $f(x) = r a^r / x^{r+1}$ for $x \ge a$, the cumulative is $F(x) = (a/x)^r$, yielding the classical result:

$$\text{pdf exponent} = \text{cumulative exponent} + 1$$

However, for variables with geometrically growing discrete support, $k \in \{ c, c m, c m^2, \ldots \}$, and $p(k) = r a^r / k^{r+1}$, the cumulative is

$F(k) = \sum_{i: k_i \geq k} p(k_i) \propto k^{-r}$

with

$\text{pdf exponent} = \text{cumulative exponent}$

This distinction is crucial in complex networks with degree sequences restricted to such geometrically growing supports (e.g., Apollonian networks), where improper application of the conventional exponent shift leads to erroneous scaling estimates. Numerical simulations confirm the analytical prediction: on log–log plots, both distribution and its cumulative show identical slopes in the geometric domain case (Guo, 2010).

3. Sharp Bounds and Inference via Localized Cumulative Distributions

Bounding cumulative probabilities—especially in tails or boundary regions—is a classical concern in statistics and signal processing. LCDs enable sharp, monotonicity-driven bounds for complex distributions. Notably, ratios of integrals are bounded by ratios of their monotonic integrands, leading to tight inequalities for noncentral gamma and beta CDFs (Segura, 2015):

• For noncentral gamma:

$$P_{\mu+1}(x, y)/P_{\mu}(x, y) < \sqrt{y/x}\, \frac{I_{\mu}(2\sqrt{xy})}{I_{\mu-1}(2\sqrt{xy})}$$

• Complementary distribution bounds via error functions or incomplete gamma functions:

$$Q_{\nu}(x, y) > \text{(functions of } \text{erfc}, \gamma_{\nu}(y), \text{etc.)}$$

These bounds facilitate practical inversion (quantile function estimation), error control in algorithms, and efficient confidence band construction in LCD-based analyses. The approach generalizes to weighted and modular forms—key for localized probability assessment in high-dimensional or stratified domains.

4. LCDs in Network Science and Scale-Free Structures

LCD concepts parallel the separation between global and localized cumulative measures in dynamic and modular networks. Definitions:

• Global cumulative distribution: $$P_{\mathrm{cum}}(k) = \frac{1}{n_v} \sum_{k' > k} N(k', t)$$
• Edge-cumulative distribution: $$P_{\mathrm{ecum}}(k) = \frac{1}{n_e} \sum_{k' > k} E(k', t)$$

Algorithmic construction of deterministic scale-free networks (recursive, Sierpinski, Apollonian) shows both cumulative measures display equivalent power-laws: $P_{\mathrm{cum}}(k) \propto k^{1-\gamma}$, and similar holds for edge-cumulative statistics (Wang et al., 2016). LCDs, restricted to local modules, are expected to inherit analogous scaling, though precise behavior under randomness or modular cutoff remains open.

These approaches allow for heterogeneity analysis, identification of local hubs, or boundary-sensitive network growth characterization. Open problems include equivalence validation in random networks and computational efficiency of edge- vs. vertex-based localizational strategies.

5. Statistical Inference and Confidence Bands Using Localized CDFs

Localized supremum inequalities generalize the canonical Dvoretzky–Kiefer–Wolfowitz (DKW) bounds for empirical CDF deviations to arbitrary subintervals. The probability

$$P \left( \sup_{u \in [\underline{u}, \overline{u}]} U_n(u) - U(u) > \epsilon \right)$$

can be exactly computed for each $n, \epsilon, [\underline{u}, \overline{u}]$, enabling finer-grained confidence bands essential for tail-sensitive risk measures (CVaR, etc.) (Odalric-Ambrym, 2020). Compared to the global Massart bound $\sqrt{\frac{\ln(1/\delta)}{2n}}$, localized bounds are substantially sharper over restricted intervals, particularly when $[\underline{u}, \overline{u}]$ is small.

Exact inversion of these relationships (for fixed confidence $\delta$) yields data-driven error widths for LCD-based risk quantification, which are critical in sequential analysis and financial decision-making scenarios. Extensions to time-uniform bounds, using reflection inequalities (James), support robust sequential strategy design.

6. Practical Algorithms and Positioning Error in Complex Sensing Systems

LCDs enable rigorous derivation of error probabilities in non-linear Center-of-Gravity (COG) positioning algorithms for tracking detectors. The cumulative probability that a COG-estimated position $x$ is less than a threshold is obtained by integration over complicated regions defined by non-linear combinations and switching rules (arising from noise-driven changes in the dominant channel):

• Two-strip: $x = \xi / (\xi + \eta)$
• Three-strip complete: $x = (x_1 - x_3)/(x_1 + x_2 + x_3)$
• Complex switching (Heaviside cases): see (Landi et al., 2021)

Transformations (e.g., $\eta', \xi'$) regularize integration domains, accommodating discontinuities induced by noise-triggered regime switches. Differentiation of these LCDs yields accurate PDFs describing non-Gaussian, heavy-tailed error distributions with gaps—phenomena critical to robust maximum likelihood track fitting and understanding detector behavior (Landi et al., 2020, Landi et al., 2021).

7. Applications in Optimization, Control, and Statistical Estimation

LCD-based sampling underpins new algorithms in optimal control and statistical inference. Notably, deterministic sampling CEM (dsCEM) utilizes pre-computed Dirac sample sets matching the LCD of a target density, enabling sample-efficient and smooth exploration in model predictive control (MPC):

$F(m, b) = \int f(x) K(x, m, b)\,dx$

Optimal sample sets are generated by minimizing modified Cramér–von Mises distance between target and Dirac mixture LCDs. Online, these samples transform to current proposals via $u^{(i)} = \mu_j + L_j \tilde{u}^{(i)}$ (Walker et al., 7 Oct 2025). Compared to stochastic Monte Carlo samples, LCD-based deterministic sets offer:

• Reduced sample budget for convergence
• Smoother control trajectories (lower jitter, actuator stress)
• Modular extension to high-dimensional control and probabilistic inference

Experimental results confirm dsCEM's superiority in cost minimization and smoothness, especially in low-sample regimes.

In econometric inference, local regression distribution estimators employ localized CDF approximations—via kernel-weighted polynomial bases—to achieve boundary-adaptive density estimation, pointwise Gaussian limit results, and efficient confidence band construction (Cattaneo et al., 2020). These facilitate robust quantile estimation, heterogeneity analysis, and program evaluation.

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Localized cumulative distributions represent a versatile and technically rigorous framework for probability quantification, statistical inference, and optimization over restricted domains or modular data structures. Their mathematical generality and practical effectiveness continue to motivate advances in both theory and diverse application areas, with ongoing research exploring new connections, computational strategies, and sharp bounds suited to complex real-world systems.