Stat.Sample­Split

Independence of folds. For a fixed sample size n, the sample sub-tuple indexed by the nuisance fold A(n) is independent, under μ, of the sub-tuple indexed by the estimation fold B(n).

Independence of evaluation fold and training complement. For a fixed sample size n and fold index k, the sample sub-tuple indexed by the evaluation fold is independent, under μ, of the sub-tuple indexed by the training complement.

Given an iid sample, a measurable category label, a finite pilot block, a positive selection threshold, a lower population-mass band, an upper population-mass band, an upper-tail exponential tilt, a lower-tail exponential tilt, nonnegativity of the upper-tail tilt, and nonpositivity of the lower-tail tilt, failure of the simultaneous category-mass sandwich has probability at most the number of categories times the sum of the two explicit Chernoff tails. The bound also covers empty blocks and empty category types.

If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, the common risk bound is nonnegative, every branch selected on that event is eligible, eligible branches have integrable squared tail error, and their fixed-branch risks obey the common bound, the pilot-selected risk on the event obeys that same bound.

Centered fold-B empirical sum is o_p(1) from L²-rate o_p(1). For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training data and the outcome, for every n, measurable with respect to the fold-A σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$ under the sampling measure, the centered fold-B empirical sum of f — the estimation-fold sample average of f n ω minus its population mean, rescaled by $\sqrt{|B(n)|}$ — is $o_p(1)$.

Bias term is o_p(1) under fixed-ratio split. Given a positive limiting fold-B sampling ratio $c$ with the fold-B fraction $|B(n)|/n$ converging to $c$, and a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$ under the sampling measure, the bias term $\sqrt{|B(n)|}\cdot\int f(n,\omega,\cdot)\,dP$ is $o_p(1)$.

Per-fold centered evaluation sum is o_p(1) (K-fold). For an i.i.d. sample and a K-fold split, fixing an evaluation fold k, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training-complement data and the outcome, for every n, measurable with respect to the training-complement σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$, the centered per-fold empirical sum of f over the evaluation fold k — its fold sample average minus its population mean, rescaled by the square root of the fold size — is $o_p(1)$.

Per-fold bias term is o_p(1) under a nonempty K-fold split. For an i.i.d. sample and a K-fold split with a positive number of folds $K$, given a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$, the per-fold bias term $\sqrt{|{\rm fold}(n,k)|}\cdot\int f(n,\omega,\cdot)\,dP$ at evaluation fold k is $o_p(1)$.

Fold-B weak law of large numbers. For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a measurable statistic g that is square-integrable under the population measure, the estimation-fold sample average of g converges in probability to the population integral $\int g\,dP$ as the sample size grows.

Fold-B CLT. Along an i.i.d. sample S under a one-shot sample split split, fix a transform ψ : X → ℝ that is measurable, has population mean zero under P, and is square-integrable; provided the fold-B normalized partial sum is almost-everywhere measurable at every sample size, the fold-B normalized partial sum (1/√|B(n)|) Σ_{i∈B(n)} ψ(Z_i) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.

Fold-B asymptotic linearity ⇒ asymptotic normality at rate √|B(n)|. Along an i.i.d. sample S under a one-shot sample split split, if the estimator sequence θn is fold-B asymptotically linear toward θ₀ with influence function ψ that is measurable, and if the rescaled estimator and the fold-B normalized influence-function sum are almost-everywhere measurable at every sample size, then √|B(n)| · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.

Conversion to √n-rate under a fixed split ratio. Along an i.i.d. sample S under a one-shot sample split split with a fold-B asymptotically linear estimator sequence θn (h), suppose the split ratio c is strictly positive, the estimation-fold share |B(n)|/n converges to c, the influence function ψ is measurable, and the √n-rescaled estimator is almost-everywhere measurable at every sample size; then √n · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance (∫ψ²dP)/c.