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An income stream with per-period variance $b^2$ and correlation $\rho$ with a Gaussian traded price increment produces a hedge holding $-\rho b/\sigma$ in addition to speculative demand. The residual income variance is $b^2(1-{\rho }^2)$. With independent period pairs, the same conditional quadratic minimization applies at each date. A deterministic income fee changes the optimized value but not the hedge holding.

With the convention $d_{\mathrm{Hell}}^2=\frac{1}{2}\int (\sqrt{p}-\sqrt{q}{)}^2$, the Hellinger distance between two normal distributions with ${\sigma }_1,{\sigma }_2>0$ isTo prove it, integrate the geometric mean of their probability density functions. Completing the square makes that geometric mean the density of a normal distribution times the displayed prefactor and exponential. For example, writing $S={\sigma }_1^2+{\sigma }_2^2$, that normal distribution has mean $({\theta }_1{\sigma }_2^2+{\theta }_2{\sigma }_1^2)/S$ and variance $2{\sigma }_1^2{\sigma }_2^2/S$. The identity $d_{\mathrm{Hell}}^2=1-\int \sqrt{pq}$ then proves the formula.

For equal unit variances, this becomes $1-e^{-({\theta }_1-{\theta }_2{)}^2/8}$. More generally, put $B=\sqrt{2{\sigma }_1{\sigma }_2/S}\le 1$ and $x=({\theta }_1-{\theta }_2{)}^2/(4S)$. Then $1-B{e}^{-x}=(1-B)+B(1-e^{-x})$, $1-B=({\sigma }_1-{\sigma }_2{)}^2/[S(1+B)]$, and $1-e^{-x}\le x$, proving

Observations or errors are heteroscedastic when their variances are not all equal.

Constancy of the error variance across observations or conditional covariate values. In fixed-design nonparametric regression, $Y_i=m(x_i)+\sigma {ϵ}_i$ with $\mathrm{Var}({ϵ}_i)=1$ has $\mathrm{Var}(Y_i)={\sigma }^2$ for every design point.

For fractional Brownian motion, $H\in (0,1)$ controls the variance $|t-s{|}^{2H}$ of increments and the scaling of its finite-dimensional distributions. Larger $H$ means smaller increment variance on intervals shorter than one and greater sample-path Hölder continuity.

For two independent rate-$\lambda >0$ exponential distributions, set $U=\min (X,Y)$ and $V=\max (X,Y)$. Summing the two ordering branches gives joint probability density $2{\lambda }^2{e}^{-\lambda (2u+d)}$ for $(U,d=V-U)$ with $u,d>0$. It factors into independent exponential probability density functions. Thus $V$ is distributed as $X+Y/2$, with expected value $3/(2\lambda )$ and variance $5/(4{\lambda }^2)$. This spacing result is the order-statistic manifestation of the memorylessness of the exponential distribution.

For independent subgroup estimates ${\hat{\mu }}_1,{\hat{\mu }}_2$ with variances $v_1,v_2$, the null contrast has Wald statistic $Z=({\hat{\mu }}_1-{\hat{\mu }}_2)/\sqrt{v_1+v_2}$. If evidence overlaps, include minus twice the covariance in the contrast variance. A nonsignificant comparison is not proof of equivalence.

For independent studies comparing treatments $A$ and $B$ to common control $C$, a consistent log odds ratio comparison uses ${\hat{\delta }}_{AB}={\hat{\delta }}_{AC}-{\hat{\delta }}_{BC}$, with variance $v_{AC}+v_{BC}$. This follows from subtracting two estimates of additive effects relative to the same reference. With overlapping evidence, subtract twice their covariance. Causal comparability across the studies is supplied by transitivity in network meta-analysis, not by the variance calculation.

An exact $P(k)=A/k^3$ has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of $r,r_0$. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.

For a random walk with independent identically distributed steps of mean zero and variance one, the Donsker invariance principle and the continuous mapping theorem for integration give this limit. The integral of the polygonal interpolation differs from the right-endpoint sum by $S_n/(2n^{3/2})$, whose squared expectation is $1/(4n^2)$. The Slutsky theorem removes that error. The limiting variable is the time-one value of integrated Brownian motion, with law $N(0,1/3)$.

At an interior point where the regression function is differentiable, a symmetric compactly supported regression kernel makes the first-order bias of an estimator vanish asymptotically. In regular fixed-design nonparametric regression with $h\to 0$ and $nh\to \infty$, the local constant estimator then has squared bias of an estimator $o(h^2)$ and variance $R(K)/(nh)+o((nh{)}^{-1})$ for a unit-integral regression kernel. The differentiability remainder is pointwise and need not hold uniformly over a function class.

An intrinsically stationary random field has constant mean and finite increment variances depending only on displacement. Its semivariogram is $\gamma (h)=\frac{1}{2}\mathrm{Var}[Z(s+h)-Z(s)]$. Second-order stationarity implies this property, but the converse can fail: subtracting the value at a fixed origin from a stationary field preserves increments and makes marginal variance vary with location.

A weight $w_i=1/v_i$ gives more weight to a more precise estimate. In random-effects meta-analysis the variance of the corresponding marginal distribution is $v_i+{\tau }^2$, giving weight $(v_i+{\tau }^2{)}^{-1}$. A percentage weight divides the individual weight by the sum of all weights.

For independent unbiased estimators $X_i$ of a common mean with known positive variances $v_i$, the linear unbiased estimator with smallest variance has weights $w_i=v_i^{-1}/{\sum }_j{v}_j^{-1}$ and variance $({\sum }_j{v}_j^{-1}{)}^{-1}$. Unbiasedness requires ${\sum }_i{w}_i=1$. Minimizing ${\sum }_i{v}_i{w}_i^2$ by a Lagrange multiplier gives the weights, and the strictly positive diagonal Hessian matrix proves a unique global minimum. This conclusion requires no normality assumption.

Kriging predicts a spatial random field by a linear combination of observations chosen to minimize prediction variance subject to the appropriate mean constraints. Its different forms depend on whether the mean is known, an unknown constant, or an unknown regression drift.

For finite fourth moment and positive variance, kurtosis is $\mathbb{E}[(X-\mathbb{E}X{)}^4]/\mathrm{Var}(X{)}^2$. The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal.