One-Dimensional Random-Set Framework

• The one-dimensional random-set framework is a method using interval-valued random sets and measurable selections to generalize classical expectation theory.
• It employs convex duality and scalar constraints to rigorously derive explicit bounds for means, medians, and higher moments.
• Applications span robust statistics, econometric partial identification, and risk analysis where data ambiguity and partial observability are critical.

A one-dimensional random-set framework generalizes classical expectation theory by considering random intervals in $\mathbb{R}$, focusing on the collection of measurable selections and their induced moments or quantiles under additional scalar constraints. The foundational structure leverages Aumann's theory of set-valued integration but restricts attention to the minimal case of interval-valued random sets over non-atomic probability spaces. This paradigm enables precise characterization of attainable expectation ranges and their refinement under mean, median, higher-moment, and quantile restrictions, with applications to bounding event probabilities and identifying sharp moment or quantile ranges under partial observability (Beresteanu et al., 4 Dec 2025).

1. Fundamental Structure: Random Intervals and Measurable Selections

Let $(\Omega,\mathcal{A},P)$ be a complete non-atomic probability space. A random interval in $\mathbb{R}$ is a measurable map

$$Y:\Omega \rightarrow \mathcal{K}(\mathbb{R}),\quad Y(\omega) = [y_L(\omega), y_U(\omega)],$$

where $y_L,y_U$ are real random variables with $y_L\leq y_U$ a.s., and $E(|y_L|+|y_U|)<\infty$. The set of measurable selections is

$\Sel(Y) = \{y:\Omega\to\mathbb{R}: y(\omega)\in Y(\omega)\ \text{a.s.}\}.$

The Aumann expectation (selection expectation) of $Y$ is defined as

$E[Y]:= \overline{\{\,E[y]:\,y\in\Sel^1(Y)\,\}}\subseteq\mathbb{R}.$

In the case of random intervals, this reduces to

$E[Y] = [E(y_L),E(y_U)],$

so the unrestricted range of possible means across all measurable selections is exactly the interval $[E(y_L),E(y_U)]$.

2. Mean-Restricted Selection Sets and Existence

For $\kappa\in\mathbb{R}$, the $\kappa$–mean restricted selection set is

$\Sel(Y\,|\,\kappa) := \{y\in\Sel(Y): E[y] = \kappa \}.$

A key result is that every $\kappa$ in the Aumann interval $[E(y_L), E(y_U)]$ is attainable; the set $\Sel(Y\,|\,\kappa)$ is nonempty for each such $\kappa$. Convex mixtures $y_t := (1-t)y_L + t y_U$ ($t\in[0,1]$) yield $E[y_t] = (1-t)E[y_L] + t E[y_U]$ and therefore cover the entire interval. No further structural assumptions beyond non-atomicity and integrability are required [(Beresteanu et al., 4 Dec 2025), Prop. 2.3].

3. Refinements: Scalar-Constraint Restricted Selections

Additional scalar constraints shrink the range of attainable means or other functionals.

3.1. Median Restrictions

Given $m\in\mathbb{R}$, the $m$–median restricted selection set is

$\Sel(Y\,|\,m) := \{y\in\Sel(Y): m\in\Med(y)\},$

where $\Med(y)$ is the set-valued population median of $y$. The range $\{E[y]:y\in\Sel(Y\,|\,m)\}$ is a compact interval $[E_{\min}(m),E_{\max}(m)]$ under feasibility conditions. Explicit formulas involve distributions of the upper/lower gaps at $m$ given $M=\{y_L\leq m\leq y_U\}$, and "thresholded" selections can attain every mean in this range [(Beresteanu et al., 4 Dec 2025), Prop. 2.8].

3.2. Higher-Moment Constraints

For real $r$ and a target moment $\mu_r$, the restricted set is

$\Sel_r(Y\,|\,\mu_r) = \{y\in\Sel(Y): E[y^r]=\mu_r\}.$

Nonemptiness holds if and only if $\mu_r\in [E(y_L^r),E(y_U^r)]$ (under mild monotonicity). The set of achievable means $\{E[y]:y\in\Sel_r(Y\,|\,\mu_r)\}$ is a compact interval admitting a dual representation: $\sup_{y\in\Sel_r(Y\,|\,\mu_r)}E[y]=\inf_{\lambda\in\mathbb{R}}\{E[\sup_{x\in Y}(x+\lambda x^r)]-\lambda \mu_r\},$ with a symmetric formula for the infimum. The maximization exchanges selection pointwise and in expectation.

3.3. General Quantile Restrictions

For $\alpha\in(0,1)$ and quantile $q_\alpha$, the class

$\Sel_\alpha(Y\,|\,q_\alpha) = \{y\in\Sel(Y): F_y^{-1}(\alpha)=q_\alpha\}$

is nonempty if $q_\alpha\in[T_Y^{-1}(\alpha), C_Y^{-1}(\alpha)]$ (capacity and containment quantiles). The induced mean range $\Theta_E(\alpha,q_\alpha)$ is a convex compact interval $$[\underline m(\alpha,q_\alpha),\overline m(\alpha,q_\alpha)]$$.

4. Event Probability Bounds and Dual Representations

For Borel $A\subseteq\mathbb{R}$, the probability $P\{y\in A\}$ for $y\in\Sel(Y)$ satisfies

$$P\{Y\subseteq A\} \leq P\{y\in A\} \leq P\{Y\cap A\neq\varnothing\},$$

and under mean restriction $E[y]=\kappa$,

$\sup_{y\in\Sel(Y\,|\,\kappa)}P\{y\in A\} = \inf_{\lambda\in\mathbb{R}}\{E[\Psi(\lambda)]-\lambda\kappa\},$

where $\Psi(\lambda,\omega)$ is defined in terms of the extremal points of $Y(\omega)\cap A$ and the sign of $\lambda$. Calibrated threshold selections can attain these bounds in closed form [(Beresteanu et al., 4 Dec 2025), Thms. 3.3–3.4, Cor. 3.5].

5. Illustrative Examples

The framework accommodates explicit examples elucidating the structure of selection sets:

• Two-state symmetric interval: If $Y(\omega_1)=[\kappa-2d,\kappa]$, $Y(\omega_2)=[\kappa,\kappa+2d]$ (each with $P=1/2$), $E(y_L)=E(y_U)=\kappa$, and any median in $[\kappa-2d,\kappa]$ can be realized with $E[y]=\kappa$. Mean restrictions alone do not always reduce the attainable median set.
• Chi-square bound: For $y_L=F^{-1}_{\chi^2_2}(U)$, $y_U=F^{-1}_{\chi^2_5}(U)$ with $U\sim\mathrm{Unif}(0,1)$, and a fixed feasible median $m$, the attainable mean interval can be computed explicitly via integral formulas, and extremal selections correspond to truncating at calibrated quantiles [(Beresteanu et al., 4 Dec 2025), Exs. 2.4, 2.7].

6. Generalizations and Outlook

Extensions include higher-moment constraints via two-dimensional random sets $\Xi(\omega)=\{(x,x^r):x\in Y(\omega)\}$, employing convex duality for sharp bounds. Similarly, general quantile constraints $\alpha\neq 1/2$ yield mean intervals $$[\underline{m}(\alpha,q_\alpha),\overline{m}(\alpha,q_\alpha)]$$ for each admissible $(\alpha,q_\alpha)$. Both cases benefit from Hardy–Littlewood–Pólya rearrangement arguments, producing closed-form dual formulas and explicit construction of extremal measurable selections. These techniques support advanced identification analysis in models with partially observed or interval-valued latent variables (Beresteanu et al., 4 Dec 2025).

7. Significance in Applied and Theoretical Contexts

The one-dimensional random-set framework establishes a minimal, analytically tractable foundation for handling ambiguity and partial identification in random interval models. The exact attainment results for mean, moment, and quantile ranges under scalar constraints play a central role in robust statistics, econometric partial identification, and risk analysis where only range-valued data or latent structures are observed. The convexity and duality principles underlying mean and probability bounds provide a unified approach to sharp identification and optimal selection under incomplete information (Beresteanu et al., 4 Dec 2025).

Restriction	Nonemptiness Condition	Attainable Range Structure
None (unrestricted)	always	$[E(y_L),\,E(y_U)]$
Mean $E[y]=\kappa$	$\kappa\in[E(y_L),E(y_U)]$	$\{\kappa\}$
Median $m$	$m$ between capacity/containment medians	$[E_{\min}(m), E_{\max}(m)]$
Moment $E[y^r]=\mu_r$	$\mu_r\in[E(y_L^r), E(y_U^r)]$	$[m_{\min}(\mu_r), m_{\max}(\mu_r)]$
Quantile $q_\alpha$	$q_\alpha\in[T_Y^{-1}(\alpha), C_Y^{-1}(\alpha)]$	$$[\underline{m}(\alpha,q_\alpha), \overline{m}(\alpha,q_\alpha)]$$

A plausible implication is that this paradigm provides a universal bounding device for moment and quantile identification under interval uncertainty, given only minimal probabilistic structure and continuity.