Gaussian concentration and Lipschitz functions

Let Z be a standard Gaussian vector in R^d, with covariance I_d, and let f be globally L-Lipschitz for Euclidean distance. Then f(Z) is integrable, and its centered log mgf is at most L²λ²/2 for every real λ. Consequently each directed deviation above or below its mean has bound exp(−t²/(2L²)); their union has factor two. The constant-function case is deterministic. The center here is the expectation, not an interchangeable median with unchanged constants.

A differentiable function with gradient norm at most L everywhere is L-Lipschitz on Euclidean space. Nondifferentiable functions such as the Euclidean norm or sums of absolute values can instead be checked directly using triangle and Cauchy–Schwarz inequalities. A bound on each coordinate derivative by one gives a possible Euclidean gradient bound sqrt(d), not one. Nor does a function become globally Lipschitz merely because its derivatives are small on a plotted interval or on a high-probability region; extending a local claim requires additional work.

For a Gaussian vector X=m+Σ^(1/2)Z, composition with the linear covariance map multiplies the Lipschitz constant by at most sqrt of the operator norm of Σ. Thus the variance proxy becomes L² times that operator norm. This includes degenerate covariance matrices; if the resulting proxy is zero, the observable is constant almost surely under the supported directions. Replacing a standard Gaussian by one with covariance I_d/d divides the proxy by d, so ignoring normalization loses exactly the dimension factor one might be trying to study.

For the average of d coordinate absolute values, the Euclidean Lipschitz constant is one over sqrt(d), although each summand alone has constant one. Its mean stays sqrt(2/π), while its fluctuations shrink as d grows. By contrast the Euclidean norm has constant one, and its mean grows on the order of sqrt(d). Both satisfy the same concentration principle. The different locations and scales come from the observable, not from contradictory statements about dimension-free behaviour.

The output f(Z) generally is not normally distributed. Its centered mgf has a Gaussian upper envelope, which is enough for tails but does not identify its density or exact quantiles. A maximum of Gaussian coordinates, a norm, and an absolute-value average all have different output laws. One should therefore avoid using this inequality as if it supplied an exact normal standard error for an arbitrary nonlinear statistic. The guarantee is about deviation probability under the stated input law and global sensitivity.

The squared norm shows a useful boundary. Its gradient is twice the input vector and is unbounded, so a global fixed Lipschitz constant is unavailable. It has its own chi-square and subexponential concentration inequalities, rather than the direct theorem above. Applying the Lipschitz theorem to the norm first can still yield useful squared-norm statements after translating the threshold carefully. For a one-sided upper threshold, an upper bound on the norm's mean can help; that same mean bound does not automatically give a matching lower-tail statement.