Moment Concentration Bound

Definition

A moment concentration bound is a finite-sample guarantee that quantifies how far the estimated
mean and covariance of a distribution — learned from $N$ samples — can deviate from the true moments,
with a chosen confidence. In ren2022chance (Theorem 1) it is used to
robustify a deterministically-reformulated chance / CVaR constraint against moment-estimation error:
the true (unknown) moments $({\mu }_k,{\Sigma }_k)$ of each Gaussian-mixture mode are replaced by estimates
$({\hat{\mu }}_k,{\hat{\Sigma }}_k)$ inflated by additive (mean) and multiplicative (covariance) margins, so a
solution feasible for the inflated constraint is feasible for the exact one with probability at least
$1-2\beta$. The construction is distribution-/regime-agnostic (it is a property of the sample statistics,
not of any robot); the source’s application is automotive trajectory planning, not a space robot — neither
free-flying nor free-floating.

Key Equations

Symbols per notation.md.

Notation flag. The source writes the deterministic-reformulation coefficient as ${\Gamma }_k$
(inverse-CDF / CVaR coefficient). The glyph $\mathbf{\Gamma }$ in notation.md is the
load-bearing $12\times (6+n)$ circumcentroidal coordinate transform — an unrelated object. To avoid
collision this page renames the source’s scalar to $c_k$. The sample size $N_k$ and mode index $k$ are
local to this page (not in the registry). Two further glyphs do collide with the registry and are used
here only in the source’s risk-layer sense: the safety tolerance $\beta \in (0,1)$ here is not
notation.md’s $\beta$ (the isotropic damping floor), and the per-mode risk bound ${ϵ}_k$ (scalar) is
unrelated to notation.md’s quaternion vector part $\mathbf{ϵ}$.

Robustified per-mode constraint (feasible $\Rightarrow$ exact constraint holds w.p. $\ge 1-2\beta$):

$$c_k\sqrt{(1+r_{2,k}){\tilde{\mathbf{x}}}^{\top }{\hat{\mathbf{\Sigma }}}_k\tilde{\mathbf{x}}}+r_{1,k}(x)+{\hat{\mathbf{\mu }}}_k^{\top }\tilde{\mathbf{x}}\le 0,\forall k\in \{1,\dots ,K\},$$

with the mean margin (additive, one-sided, exploiting the scalar linear-constraint structure
${\delta }^{\top }\tilde{\mathbf{x}}$) and the covariance margin (multiplicative inflation factor):

$$r_{1,k}(x)=\sqrt{\frac{T_{1,N_k-1}^2(1-\beta )}{N_k}}\sqrt{{\tilde{\mathbf{x}}}^{\top }{\hat{\mathbf{\Sigma }}}_k\tilde{\mathbf{x}}},r_{2,k}=\max \left\{\left|1-\frac{N_k-1}{{\chi }_{N_k-1,1-\beta /2}^2}\right|,\left|1-\frac{N_k-1}{{\chi }_{N_k-1,\beta /2}^2}\right|\right\},$$

where $T_{a,b}^2(p)$ is the $p$-quantile of Hotelling’s $T^2$-distribution and ${\chi }_{k,p}^2$ the
$p$-quantile of the ${\chi }^2$-distribution. The coefficient is $c_k={\Psi }^{-1}(1-{ϵ}_k)$ for a chance
constraint and $c_k=\Phi ({\Psi }^{-1}(1-{ϵ}_k))/{ϵ}_k$ for its CVaR approximation
(${\Psi }^{-1},\Phi$ = inverse-CDF / PDF of $\mathcal{N}(0,1)$). The mean margin $r_{1,k}$ (1-D, $n=1$) is
no looser than the multivariate ball bound $R_{1,k}$ of prior work — the source’s Remark 2 proves
${\lambda }_{\max }(\hat{\mathbf{\Sigma }}){\tilde{\mathbf{x}}}^{\top }\tilde{\mathbf{x}}\ge {\tilde{\mathbf{x}}}^{\top }\hat{\mathbf{\Sigma }}\tilde{\mathbf{x}}$
(a $\ge$, not a strict inequality) and reports the constant gap $C_1\ge C_2$ as an empirical
observation for $n\ge 1$, concluding only that the 1-D bound “appears to be tighter.”

Source Support

• ren2022chance — primary and sole source: derives the GMM moment
  concentration bound (Theorem 1), splits it into a Hotelling-$T^2$ mean margin and a ${\chi }^2$ covariance
  inflation, and uses it to certify finite-sample feasibility of chance and CVaR constraints in
  trajectory planning.

Related Topics

• chance_constraints — the bound robustifies a chance constraint against
  moment-estimation error, preserving the $1-ϵ$ violation guarantee under learned moments.
• conditional_value_at_risk — the same bound certifies the CVaR
  approximation of the chance constraint (the coefficient $c_k$ switches to the CVaR form).
• covariance_propagation — supplies the covariance ${\hat{\mathbf{\Sigma }}}_k$
  that the ${\chi }^2$ factor $r_{2,k}$ inflates; this bound quantifies the trust placed in that estimate.
• interval_arithmetic — an alternative set-based way to carry uncertainty
  margins through a constraint; contrast with this distributional, sample-count-driven margin.
• coherent_risk_measures — CVaR (the approximation certified here) is the
  canonical coherent risk measure; the bound is what makes its finite-sample use rigorous.

Open Questions

• The source is automotive (linear time-varying ego vehicle, GMM obstacle uncertainty). Does the bound
  transfer to our free-flying manipulator’s chance-constrained planning, where the constraint map
  ${\delta }^{\top }\tilde{\mathbf{x}}$ is replaced by a nonlinear function of the coupled base+arm state?
• The $1-2\beta$ guarantee assumes the moments are estimated by independent sampling per GMM mode. For
  on-orbit estimation (e.g. recursive filter outputs), the i.i.d.-sample premise behind the
  Hotelling-$T^2$ / ${\chi }^2$ quantiles may not hold — what replaces $N_k$?
• The bound assumes each mode is Gaussian (${\delta }^{\top }\tilde{\mathbf{x}}$ Gaussian). What is the
  analogous concentration margin when the per-mode model is itself only approximate?