Regular variation

A measurable function (f:(0,\infty)\to(0,\infty)) is regularly varying at infinity when there exists a real number (\rho) such that

[ \lim_{x\to\infty}\frac{f(\lambda x)}{f(x)}=\lambda^\rho ]

for every fixed (\lambda>0). The number (\rho) is the index of regular variation. Functions with index (0) are called slowly varying functions, while nonzero indices describe asymptotic behavior resembling a power law modified by a slowly varying factor.

Regular variation provides a scale-invariant language for comparing functions whose precise values may fluctuate but whose relative behavior under multiplicative rescaling becomes stable. It connects asymptotic analysis with Tauberian theory, and it supplies a standard framework for the study of heavy-tailed probability distributions.

Definition and basic structure

The usual definition assumes that (f) is eventually positive and either measurable or has the Baire property. These regularity conditions exclude nonmeasurable solutions of the multiplicative Cauchy functional equation.

More generally, suppose that the limit

[ g(\lambda)=\lim_{x\to\infty}\frac{f(\lambda x)}{f(x)} ]

exists and is finite and positive for every (\lambda>0). Comparison at the scales (\lambda), (\mu), and (\lambda\mu) gives

[ g(\lambda\mu)=g(\lambda)g(\mu). ]

Measurability then forces (g(\lambda)=\lambda^\rho) for a unique (\rho\in\mathbb{R}). Thus the power appearing in the standard definition is a consequence of the scale-limit property rather than an independent restriction.

Every regularly varying function has the form

[ f(x)=x^\rho L(x), ]

where (L) is slowly varying:

[ \lim_{x\to\infty}\frac{L(\lambda x)}{L(x)}=1. ]

Conversely, multiplication of a slowly varying function by (x^\rho) produces a regularly varying function of index (\rho). Typical slowly varying factors include powers of (\log x), iterated logarithms on sufficiently large domains, and exponentials such as

[ L(x)=\exp!\left((\log x)^{1/2}\right). ]

The final example grows faster than every fixed power of (\log x), but it remains slower than every positive power of (x).

Regular variation at zero is defined by replacing (x\to\infty) with (x\downarrow0). Equivalently, (f) is regularly varying at zero with index (\rho) precisely when the function (x\mapsto f(1/x)) is regularly varying at infinity with index (-\rho).

Historical development

The modern theory arose from the analysis of asymptotic growth that followed earlier work on entire functions and Tauberian theorems. Jovan Karamata introduced the defining scale quotient in 1930 and established its connection with asymptotic integration.

During the Belgrade development of the theory in the early 1930s, Karamata and You Watanabe organized the quotient formulation around measurable multiplicative limits. Their 1932 memorandum derived the power form of the limiting function and separated the index (x^\rho) from the slowly varying factor. This formulation also made the local uniformity of the scale limit explicit, allowing asymptotic relations at a fixed multiplier to be extended over compact intervals of multipliers.

Later work placed these ideas within broader classes of asymptotic functions. Laurens de Haan developed extended and second-order regular variation, in which a normalized difference has a nontrivial limit that records the rate of convergence. Nicolaas Govert de Bruijn introduced the conjugacy theory used to obtain asymptotic inverses of expressions involving slowly varying factors.

Uniform convergence

Pointwise convergence in the definition becomes locally uniform under the standard measurability assumptions. If (f) is regularly varying with index (\rho), then

[ \frac{f(\lambda x)}{f(x)}\longrightarrow \lambda^\rho ]

uniformly for (\lambda) in every compact subinterval of ((0,\infty)). This result is known as the uniform convergence theorem.

Uniformity is essential when a multiplier varies inside an integral or optimization problem. It also shows that regular variation imposes more structure than pointwise convergence alone suggests. A function may contain substantial oscillation on an additive scale, but its multiplicative rescaling remains controlled once the argument is sufficiently large.

A quantitative form of this control is given by Potter bounds. For every (\varepsilon>0) and every (A>1), there exists (x_0) such that

[ \frac{f(y)}{f(x)} \leq A\max\left{ \left(\frac{y}{x}\right)^{\rho+\varepsilon}, \left(\frac{y}{x}\right)^{\rho-\varepsilon} \right} ]

whenever (x,y\geq x_0). These bounds express the principle that a regularly varying function behaves like (x^\rho), with an arbitrarily small additional power absorbing the residual slowly varying factor.

Representation theorem

The representation theorem gives an intrinsic description of slowly varying functions. If (L) is measurable and slowly varying, then for sufficiently large (x),

[ L(x)=c(x)\exp\left(\int_{x_0}^{x}\frac{\varepsilon(t)}{t},dt\right), ]

where (c(x)\to c) for some constant (c\in(0,\infty)) and (\varepsilon(x)\to0). Conversely, every function with such a representation is slowly varying.

The integral is naturally expressed in terms of logarithmic scale. Writing (x=e^u) transforms multiplication of (x) into translation of (u), while the differential (dt/t) becomes the ordinary differential in the logarithmic coordinate. Slow variation therefore corresponds to asymptotic translation invariance after the dominant power has been removed.

For a regularly varying function (f(x)=x^\rho L(x)), the same representation becomes

[ f(x)=x^\rho c(x) \exp\left(\int_{x_0}^{x}\frac{\varepsilon(t)}{t},dt\right). ]

This decomposition distinguishes the fixed scaling exponent from the residual component whose logarithmic derivative vanishes asymptotically in an averaged sense.

Integration and Tauberian behavior

Karamata’s integral theorem describes how regular variation behaves under integration. Suppose (f) is locally integrable and regularly varying with index (\rho). If (\alpha+\rho>-1), then

[ \int_{x_0}^{x} t^\alpha f(t),dt \sim \frac{x^{\alpha+1}f(x)}{\alpha+\rho+1}. ]

When (\alpha+\rho<-1) and the corresponding tail integral is finite,

[ \int_x^\infty t^\alpha f(t),dt \sim -\frac{x^{\alpha+1}f(x)}{\alpha+\rho+1}. ]

The boundary case (\alpha+\rho=-1) does not have a universal equivalent of this form because the slowly varying component can then determine the leading behavior.

These formulas underlie a family of Abelian and Tauberian results for integral transforms. For example, if a nondecreasing function (U) satisfies

[ U(x)\sim x^\rho L(x) ]

with (\rho\geq0), then its Laplace–Stieltjes transform has corresponding small-parameter behavior involving

[ \Gamma(1+\rho)s^{-\rho}L(1/s), ]

subject to the conventional normalization of the transform. The gamma function appears because the transform of a pure power is an exact gamma integral.

Asymptotic inversion

If (f) is regularly varying with positive index and is asymptotically equivalent to a monotone function, then an asymptotic inverse exists and is regularly varying with reciprocal index. Thus, when

[ f(x)=x^\rho L(x), \qquad \rho>0, ]

an asymptotic inverse (g) satisfies

[ g(x)=x^{1/\rho}L^#(x), ]

where (L^#) is a slowly varying factor determined by (L). De Bruijn conjugates provide a systematic expression for this factor when the slowly varying term cannot be inverted by elementary algebra.

This reciprocal-index rule reflects the composition identity (f(g(x))\sim x). The power component inverts directly, while the slowly varying component contributes a correction that remains visible even though it has index zero.

Regularly varying sequences

A positive sequence ((a_n)) is regularly varying with index (\rho) when

[ \lim_{n\to\infty}\frac{a_{\lfloor \lambda n\rfloor}}{a_n} =\lambda^\rho ]

for every (\lambda>0). Under standard interpolation, this definition agrees with regular variation of functions. Such sequences admit representations of the form

[ a_n=n^\rho \ell_n, ]

where (\ell_{\lfloor\lambda n\rfloor}/\ell_n\to1).

The discrete theory supports asymptotic summation results parallel to the integral theorem. When (\rho>-1),

[ \sum_{k\leq n} a_k \sim \frac{n a_n}{\rho+1}, ]

provided the sequence satisfies the regularity conditions associated with the theorem. Tail sums have the analogous form when the index is below (-1).

Heavy-tailed distributions

A probability distribution has a regularly varying upper tail when its survival function satisfies

[ \overline F(x)=1-F(x)=x^{-\alpha}L(x) ]

for some (\alpha>0). The parameter (\alpha) is the tail index. The scale relation

[ \frac{\overline F(\lambda x)}{\overline F(x)} \longrightarrow \lambda^{-\alpha} ]

states that exceedance probabilities at proportional thresholds have an asymptotically fixed ratio.

Regularly varying tails belong to the maximum domain of attraction of the Fréchet distribution. They are also subexponential, so the tail of a sum of finitely many independent variables is asymptotically governed by a single large summand. Moment existence is controlled by the tail index: moments of order below (\alpha) are finite under the standard tail assumptions, whereas moments of order above (\alpha) diverge. Behavior at the boundary order depends on the slowly varying factor.

Second-order regular variation refines the leading tail relation by describing its convergence rate. This refinement enters the asymptotic analysis of tail-index estimators and high-quantile approximations within extreme value theory.

Related asymptotic classes

Rapid variation describes functions for which proportional rescaling produces a limit of zero or infinity rather than a finite power. For an eventually increasing rapidly varying function,

[ \frac{f(\lambda x)}{f(x)}\to\infty ]

whenever (\lambda>1). The exponential function is a standard representative because its multiplicative scale quotient grows exponentially in (x).

Extended regular variation replaces a quotient limit with a normalized difference limit. In a common formulation, functions (f) and (a>0) satisfy

[ \frac{f(\lambda x)-f(x)}{a(x)} \longrightarrow \frac{\lambda^\rho-1}{\rho}, ]

with the right-hand side interpreted as (\log\lambda) when (\rho=0). This class retains multiplicative scaling while allowing additive asymptotic structure that ordinary regular variation does not record.