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Write $X_{-i}$ for all coordinates except $X_i$, let $X_i^{\prime }$ be an independent random variable with the same distribution as $X_i$, and let $Z_i^{\prime }=f(X_1,\dots ,X_i^{\prime },\dots ,X_n)$. Three equivalent forms of the Efron–Stein inequality arefor arbitrary square-integrable $Z_i$ measurable with respect to $X_{-i}$, andThe first is the sharp choice within the second because conditional expectation is the least-squares projection. The first and third right sides are equal because two conditionally independent copies have expected squared difference twice their conditional variance.

The vector $(y_1,y_2,y_3)$ is Jointly Gaussian. Assuming $R>0$, Gaussian conditional independence givesThus the required condition is $R{R}_{13}=R_{12}{R}_{23}$. Under it, conditioning on $y_1$ supplies no further information after $y_2$, and the Gaussian process regression posterior isBoth the conditional expectation and conditional variance depend only on $y_2,t_2,t_3$; neither contains $y_1$ or $t_1$.

Write $X_{-i}$ for all inputs except $X_i$. Under the Efron–Stein inequality hypotheses, variance tensorization is the equivalent conditional variance formConditioning freezes the other inputs. Then $Z,Z_i^{\prime }$ are independent with the same conditional probability distribution, givingTaking expected values explains the factor $1/2$.^[ref]

For Bernoulli input $X_i$ with parameter $p_i$, let ${\Delta }_{i}f$ be the value of $f$ with coordinate $i$ set to one minus its value with that coordinate set to zero, keeping $X_{-i}$ fixed. The two-point conditional variance isFor the maximum example, ${\Delta }_{1}f=1-X_2$ and ${\Delta }_{2}f=1-X_1$. The two averaged conditional variances are $0.3(0.7)(0.4)=0.084$ and $0.6(0.4)(0.7)=0.168$, precisely half the respective resampling terms. In the dependent example $X_1=X_2=B$, each other coordinate already determines $W$, so both conditional variances vanish; variance tensorization also fails without independence.

For a short induction proof, separate the last input using the law of total variance. Apply the smaller-dimensional statement to the conditional expectation; Jensen's inequality for the square bounds its coordinate contributions by the original ones. One input gives equality, starting the induction.^[ref]

Volatility clustering means that large absolute observations tend to be followed by further large absolute observations, and small ones by small ones. In an ARCH process, persistence is in the conditional variance rather than necessarily in the signed observations.