translationinvariant

Translation invariance is the property of remaining unchanged under translations, i.e., shifts by a vector in space or time. In strict terms, a function f: R^n -> R is translation-invariant if f(x + a) = f(x) for all x and all a; this is only possible for constant functions, so in practice the phrase is used for measures, processes, or systems whose behavior or distribution does not depend on absolute position.

In measure theory, a measure μ on a group like R^n is translation-invariant if μ(A) = μ(A + a) for all measurable A and all a. The standard Lebesgue measure is translation-invariant, and Haar measure generalizes this concept to more abstract groups.

In probability and statistics, a random field or stochastic process is said to be stationary (translation-invariant) if its joint distributions are invariant under translations: the distribution of (X(t1), ..., X(tk)) is the same as that of (X(t1 + h), ..., X(tk + h)) for all h. This means statistical properties depend only on relative locations, not on absolute time or space.

In signal processing and systems theory, a system is time-invariant (translation-invariant) if a time shift of the input yields an identical time shift of the output: T{x(t − τ)} = y(t − τ). Convolution with a fixed kernel is a canonical translation-invariant operation, leading to impulse-response representations.

Translation invariance is a fundamental symmetry in physics and mathematics, underpinning homogeneous spaces, lattice structures, and, via Noether’s theorem, conservation laws such as momentum.