Stat.Concentration.Bounded­Variation

If a continuous path has bounded variation, then it has a continuous nondecreasing cumulative-variation control that starts at zero, ends at the path's total variation, and dominates every ordered increment of the path.

If a continuous scalar control u is nondecreasing, zero at time zero, and dominates the summed squared increments of the paths over every interval from s to t, then the Rademacher energy of the paths is at most 4096 times the total of the sum of their squared values at time zero and the terminal control value.

Suppose the control u is nondecreasing, zero at time zero, and dominates the summed squared path increments over every interval, and the dyadic grids place their i-th level-k point at control value i/2^k times the terminal value and assign the same time to equal dyadic fractions at different levels. If the maximal signed increments over the levels are summable, then the absolute σ-signed path sum at any time s is at most the absolute σ-signed sum at time zero plus the total of the maximal signed increments over all levels.

For m + 1 coefficient vectors in ℝ^n, the sign-average of the largest squared Rademacher sum is at most 32 (1 + log(m + 1)) times the largest squared Euclidean norm of the vectors.

Under a probability measure, take finitely many coefficient paths c_a, random scalar coefficients X_a, and a clip radius with the radius nonnegative, every coefficient path of finite total variation, and the scalar coefficients measurable and square integrable. Then the squared size of the clipped path t ↦ clip(Σ_a X_a·c_a(t)) is integrable, and its expectation is at most (Σ_a size(c_a)²)·(Σ_a E[X_a²]), where the size of a path is its supremum norm plus its total variation.

Under a probability law, measurable nonnegative auxiliary weights that sum to one, measurable continuous paths, and an integrable weighted squared-path cost, the expected squared supremum norm of the weighted average path Σ_a w_a(x)·Z_a(x) is at most the expected weighted average Σ_a w_a(x)·‖Z_a(x)‖² of the squared supremum norms: finite conditional averaging contracts expected squared supremum risk.

If a continuous control path on the unit interval is nondecreasing and zero at time zero, then every level between zero and its terminal value is attained at some time, and these representative times can be chosen increasing in the level.

For finitely many random continuous paths that are Bochner-integrable and have integrable squared supremum norm, the expected squared supremum norm of the centered sum of the paths is at most the expected squared supremum norm of the summed differences between two independent draws of the paths.

For finitely many random continuous paths that are measurable, have integrable squared supremum norm, and are mutually independent, the expected squared supremum norm of the summed differences between two independent draws equals the expected Rademacher energy (sign-averaged squared supremum norm) of those differences.

For finitely many measurable and mutually independent random continuous paths, swapping the two independent draws in any selected coordinates leaves the joint law of the array of differences between the draws unchanged.

If a scalar control path dominates the summed squared increments of n paths over every ordered pair of times and the control takes the same value at times s and t, then every one of the paths takes the same value at s and t.

If every one of the m + 1 consecutive increments of a finite time grid has summed squared path increments at most δ, then the sign-averaged largest squared signed increment over the grid is at most 32 (1 + log(m + 1)) δ.

If the path f has finite total variation and so does the path g, then the supremum-plus-variation size of f − g is at most the sum of the sizes of f and g.

For finitely many real functions on the sign patterns of length n, the uniform sign-average of the square of their sum is at most the square of the sum of their individual root sign-average second moments.

For finitely many random continuous paths on the unit interval that are measurable, Bochner-integrable, almost surely of bounded variation, with integrable squared supremum-plus-variation size, and mutually independent, the expected squared supremum norm of their centered sum is at most 16384 times the sum of their expected squared sizes.

If each of finitely many continuous paths has bounded variation, then their Rademacher energy is at most 4096 times the sum of their squared supremum-plus-total-variation sizes.

The sum over all levels k of the square roots of 32 (1 + log 2^(k+1)) / 2^(k+1) is at most 40.

Suppose a scalar control u dominates the summed squared path increments over every interval, and the dyadic grids are monotone at every level and place their i-th level-k point at control value i/2^k times the terminal value. Then at every level k the sign-average of the squared maximal signed increment is at most 32 (1 + log 2^(k+1)) times u(1)/2^(k+1).

If, for a series of real functions on the sign patterns of length n, the root sign-average second moments of its terms are summable, then the sign-average square of the series is at most the square of the sum of those root second moments.

For a nonnegative threshold u and a coefficient vector with positive sum of squares, the fraction of sign vectors whose signed sum has absolute value at least u is at most 2·exp(−u^2/(2 times the sum of squared coefficients)), uniformly in the number of signs.

Let a time-indexed family of four-variable real polynomials and a nonnegative number B be given, with degree at most D in each variable at every time. If, at every point of the cube [−1, 1]^4, the value of the polynomial is continuous in time, has bounded variation in time, and has supremum-plus-variation size at most B, then there are continuous coefficient paths, one for each multi-index in {0, …, D}^4, such that at every time the polynomial equals the corresponding combination of products of first-kind Chebyshev polynomials, every coefficient path has bounded variation, and every coefficient path has size at most 16 B.

Let a time-indexed family of four-variable polynomials have degree at most 2(K − 1) in each variable and grid evaluations continuous in time. If its evaluation at every point of the normalized cube is continuous in time, of bounded variation, and of path size at most B, where B is nonnegative, then the summed path sizes of its monomial coefficient paths are at most 16 (2(K − 1) + 1)^4 (1 + √2)^(8(K − 1)) B.

Let a probability measure, a family of continuous paths indexed by its sample space together with a real weight function, and nonnegative numbers B and M be given. If at each time the weighted path value is integrable, the weight is bounded by M in absolute value everywhere, every path has bounded variation, and every path has supremum-plus-variation size at most B, then there is a continuous path whose value at each time is the integral of the weighted path values, which has bounded variation and size at most M B.