1 Definition and basic properties
1.1 Formal definition
A function \(L:(0,\infty)\to(0,\infty)\) is called *slowly varying at infinity* if, for every \(c>0\), \[ \lim_{x\to\infty}\frac{L(cx)}{L(x)}=1. \] If the same condition holds as \(x\to 0^+\), then \(L\) is slowly varying *at zero*. In the common asymptotic regime, \(L\) changes so gradually that multiplying the argument by a fixed constant leaves the function essentially unchanged in the limit.
1.2 Equivalent characterizations
Several formulations capture the same asymptotic idea. A typical equivalent viewpoint is that logarithmic changes in the argument translate into negligible changes in the function at the level of ratios. Concretely, the defining limit implies that for each fixed \(c\), \[ \log L(cx)-\log L(x) \to 0 \quad (x\to\infty), \] whenever \(L\) is positive so that \(\log L\) is well defined. Conversely, under mild regularity assumptions, this “vanishing log-difference” criterion can be used to recover the ratio limit.
Another useful characterization is formulated in terms of normalized increments: the function behaves like a “zeroth-order” term compared with power functions. In regular variation theory, this is expressed by stating that slowly varying functions are exactly the non-power-law component in regularly varying functions.
1.3 Basic examples
- Constant functions: \(L(x)=A\) with \(A>0\) satisfy \(L(cx)/L(x)=1\).
- Logarithmic factors: \(L(x)=(\log x)^\alpha\) for any real \(\alpha\) is slowly varying at infinity because
\[ \frac{(\log(cx))^\alpha}{(\log x)^\alpha}\to 1. \]
- Iterated logarithms: Functions such as \(L(x)=\log\log x\) (for large enough \(x\) so it is positive) are slowly varying.
- Exponential of a vanishing term: If \(L(x)=\exp\big(\varepsilon(x)\big)\) where \(\varepsilon(x)\to 0\) as \(x\to\infty\), then \(L\) is slowly varying, provided the convergence is slow enough to preserve the ratio condition.
1.4 Non-examples
- Power functions with nonzero exponent: \(L(x)=x^\beta\) yields
\[ \frac{(cx)^\beta}{x^\beta}=c^\beta\neq 1 \] for \(c\neq 1\), so \(x^\beta\) is slowly varying only when \(\beta=0\).
- Exponential growth in a power: For \(L(x)=e^{x^\gamma}\) with \(\gamma>0\),
\[ \frac{e^{(cx)^\gamma}}{e^{x^\gamma}}=e^{(c^\gamma-1)x^\gamma} \] diverges (or collapses), violating the ratio limit.
- Oscillations that do not dampen: Certain oscillatory functions can fail the limit because the ratio \(L(cx)/L(x)\) does not settle near \(1\). In contrast, oscillations are sometimes compatible with slow variation when they average out in the ratio limit (see §3.3).
2 Fundamental theory
2.1 Relation to regular variation
A function \(f\) is *regularly varying* with index \(\rho\in\mathbb{R}\) if \[ \lim_{x\to\infty}\frac{f(cx)}{f(x)}=c^\rho \quad (c>0). \] When this holds, \(f\) can be decomposed (heuristically and, under standard assumptions, rigorously) as \[ f(x)=x^\rho L(x), \] where \(L\) is slowly varying. Thus slowly varying functions serve as the “envelope” that modifies a pure power law without changing the power index.
2.2 Potter bounds
A foundational quantitative estimate is provided by Potter’s bounds. Roughly speaking, for a slowly varying \(L\) and any \(\varepsilon>0\), the function behaves like \(x^{\pm \varepsilon}\) when the argument is rescaled, uniformly away from the limit point. A standard form states: for each \(\varepsilon>0\) and each \(A>1\), there exists \(x_0\) such that for all \(x\ge x_0\) and all \(t\in[1/A,A]\), \[ \frac{L(tx)}{L(x)} \le t^\varepsilon \quad\text{and}\quad \frac{L(tx)}{L(x)} \ge t^{-\varepsilon}. \] These bounds are instrumental in establishing convergence of integrals and swapping limits with operations.
2.3 Uniform convergence theorem
While the definition uses pointwise convergence in \(c\), many asymptotic arguments require uniform control on compact sets of scaling factors. The uniform convergence theorem provides exactly this: if \(L\) is slowly varying, then \[
| \sup_{c\in[a,b]}\left | \frac{L(cx)}{L(x)}-1\right | \to 0 |
|---|
\quad (x\to\infty) \] for any \(0<a<b<\infty\). Uniformity is crucial when limits appear inside integrals or suprema.
2.4 Representation theorem
A representation theorem describes how slowly varying functions can be expressed through a simpler auxiliary function, often via integral formulas.
2.4.1 Karamata representation
Under mild assumptions (notably measurability and positivity), a slowly varying \(L\) at infinity admits a representation of the form \[ L(x)=c(x)\exp\left(\int_{x_0}^x \frac{\eta(t)}{t}\,dt\right), \] where \(c(x)\to c\in(0,\infty)\) and \(\eta(t)\to 0\) as \(t\to\infty\). Intuitively, the exponent collects the cumulative effect of a vanishing “local slope,” while \(c(x)\) captures residual multiplicative drift that stabilizes.
2.4.2 De Bruijn conjugates
For a slowly varying function \(L\), one may define a companion function \(L^\#\) (a *de Bruijn conjugate*) that compensates for \(L\)’s growth in inversion-like asymptotics. The conjugacy relation is designed so that \[ L(x)\,L^\#\!\big(x\,L(x)\big)\to 1 \quad\text{and}\quad L^\#(x)\,L\!\big(x\,L^\#(x)\big)\to 1 \] in the appropriate limit regime. Such conjugates are central in asymptotic inversion and in determining tail quantiles.
3 Examples and constructions
3.1 Logarithmic factors
As noted in §1.3, \(L(x)=(\log x)^\alpha\) is slowly varying at infinity. More generally, any factor obtained by composing a finite number of logarithms with real exponents typically remains slowly varying, as logarithms grow more slowly than powers and behave predictably under scaling.
3.2 Iterated logarithms
Iterated logarithms such as \[ L(x)=\log\log x,\qquad L(x)=\big(\log\log x\big)^\alpha,\qquad L(x)=\log\log\log x \] (for sufficiently large \(x\) so the expressions are defined) are slowly varying. Their key property is that multiplying \(x\) by a constant adds only a bounded perturbation inside the outermost logarithm, which becomes negligible relative to the unbounded growth as \(x\to\infty\).
3.3 Slowly varying functions with oscillation
Not all slowly varying functions are monotone. Constructions can introduce bounded oscillations in such a way that the ratio \(L(cx)/L(x)\) still tends to \(1\). One common method is to build \(L\) from an exponential of an oscillatory term whose amplitude decays to \(0\), for instance using a function \(\eta(t)\to 0\) in a representation like Karamata’s. The oscillations may persist but become too weak, in the ratio sense, to disrupt slow variation.
3.4 Closure properties
3.4.1 Products and quotients
If \(L_1\) and \(L_2\) are slowly varying at infinity, then so are their products and quotients: \[ L_1(x)L_2(x)\ \text{and}\ \frac{L_1(x)}{L_2(x)} \] (with the quotient defined where \(L_2(x)\neq 0\)). This follows directly from multiplying (or dividing) the ratio limits: \[ \frac{L_1(cx)L_2(cx)}{L_1(x)L_2(x)}=\frac{L_1(cx)}{L_1(x)}\cdot\frac{L_2(cx)}{L_2(x)}\to 1. \]
3.4.2 Powers and compositions
If \(L\) is slowly varying and \(\alpha\in\mathbb{R}\), then \(L(x)^\alpha\) is slowly varying. For compositions, if \(L\) is slowly varying and \(g(x)\) satisfies \(g(x)\sim x\) (or more generally \(g(cx)/g(x)\to 1\) for fixed \(c\)), then \(L(g(x))\) remains slowly varying. Such stability under mild reparametrization makes slowly varying factors flexible in applications.
4 Asymptotic analysis
4.1 Limits involving scaling
A central utility of slow variation is the ability to replace scaled arguments in asymptotic formulas. If \(L\) is slowly varying, then for fixed \(c>0\), \[ L(cx)\sim L(x)\quad (x\to\infty), \] meaning \(L(cx)/L(x)\to 1\). This allows one to simplify expressions like \(f(x)=x^\rho L(x)\) by transferring multiplicative constants into the power term while leaving \(L\) essentially unchanged.
4.2 Integrals of slowly varying factors
Integrals often reveal the “effective” growth rate. A frequently used phenomenon is that slowly varying factors behave like constants under certain integral scalings, but with predictable corrections from the power term. For example, if \(L\) is slowly varying and \(\beta>-1\), then integrals of the type \[ \int_1^x t^\beta L(t)\,dt \] are typically asymptotic to \(\frac{x^{\beta+1}}{\beta+1}L(x)\) up to a constant factor, aligning with Karamata-type theorems. When \(\beta=-1\), logarithmic terms emerge and the analysis becomes more delicate, still governed by slow variation.
4.3 Tauberian and Abelian results
Tauberian and Abelian theorems connect asymptotic behavior of transforms with asymptotic behavior of the underlying function. Slowly varying factors frequently appear in these correspondences because they remain stable under the transform’s scaling structure. Abelian results tend to show that regular variation of the original quantity implies regular variation of the transform; Tauberian results reverse the implication under additional conditions such as monotonicity or bounded variation.
4.4 Asymptotic inversion
Inversion problems ask: given an asymptotic form for a function \(F(x)\), what is the asymptotic behavior of the inverse \(F^{-1}\)? When \(F(x)\) includes a slowly varying component, inversion produces a conjugate slowly varying factor. If \[ F(x)\sim x^\rho L(x)\quad (\rho>0), \] then solutions to \(y=F(x)\) can often be expressed in terms of \(y^{1/\rho}\) times a slowly varying correction determined by a de Bruijn conjugate of \(L\). This explains why conjugates are not merely formal but practically necessary.
5 Applications
5.1 Probability theory
5.1.1 Heavy-tailed distributions
In probability, slowly varying functions arise naturally in the study of heavy tails. A common model is a survival function (tail) of the form \[ \overline{F}(x)=\mathbb{P}(X>x)\sim x^{-\alpha}L(x), \] where \(\alpha>0\) and \(L\) is slowly varying. Such distributions are intermediate between strict power laws and exponentially decaying tails, and they capture a wide range of real-world-like behaviors in purely mathematical settings.
5.1.2 Stable laws and domains of attraction
In limit theorems, sums of independent random variables can converge to stable distributions. The domain-of-attraction conditions for stable laws are often expressible using regularly varying tails, hence incorporating slowly varying functions as the non-power-law component. This provides a systematic way to identify normalization sequences and to determine scaling limits.
5.2 Number theory
5.2.1 Prime number asymptotics
Prime-counting functions and related arithmetic sums frequently involve asymptotic estimates in which slowly varying factors appear as corrections to dominant terms. While the precise form depends on the theorem and the object studied, the general mechanism is the same: logarithmic or iterated-logarithmic corrections behave like slowly varying functions.
5.2.2 Summatory functions
Summatory functions of multiplicative or arithmetic sequences can exhibit growth rates governed by regular variation. Slowly varying terms describe how arithmetic structure modifies power-law growth, especially in the presence of logarithmic factors. These refinements are important for obtaining sharper asymptotics than leading-order power terms alone.
5.3 Analysis of algorithms
In theoretical computer science, performance measures such as expected running time or distributional tails may involve slowly varying components. For instance, distributions of costs or cache-miss-like quantities can have asymptotic forms where the dominant rate is polynomial (or nearly polynomial) but includes logarithmic corrections. Recognizing slow variation can therefore improve the fidelity of asymptotic models.
5.4 Differential and integral equations
Slowly varying factors appear in asymptotic solutions of differential and integral equations, particularly those whose coefficients or forcing terms grow or decay at rates between polynomial orders and exponential rates. In such settings, solution behavior can inherit slowly varying multipliers, producing asymptotic profiles that reflect the “almost constant under scaling” nature of \(L\).
6 Related concepts
6.1 Regularly varying functions
Regularly varying functions generalize slow variation by allowing a power-law factor. If \(f(x)=x^\rho L(x)\) with \(L\) slowly varying, then \(f\) is regularly varying with index \(\rho\). This relationship organizes a large class of asymptotic behaviors under a common framework.
6.2 Rapidly varying functions
Rapid variation refers to functions whose values change drastically under small relative rescalings. This contrasts with slow variation, where multiplicative rescaling by a constant leaves ratios close to \(1\). Rapidly varying functions typically yield asymptotic regimes dominated by their non-polynomial behavior and often require different tools than regular-variation theory.
6.3 Slow variation at zero and infinity
The definition can be applied either as \(x\to\infty\) or as \(x\to 0^+\). A function can be slowly varying in one regime but not the other. Using both perspectives is useful for analyzing problems where variables approach either extreme, such as near-singularity behavior or small-parameter limits.
6.4 Matuszewska indices
Matuszewska indices provide refinements that measure the growth rates of functions more precisely than regular variation indices alone. They quantify upper and lower logarithmic slopes in a way that can classify functions lying between slow and regular variation. Slowly varying functions correspond to a neutral position where these indices match appropriate baseline values.
7 Historical notes
7.1 Origins in asymptotic analysis
Slowly varying factors emerged as part of the broader effort to formalize asymptotic comparisons beyond simple power laws. Classical asymptotic analysis often introduced “correction terms” like logarithms; regular variation provided a systematic language to treat these corrections as part of a coherent family of functions.
7.2 Contributions by Karamata and successors
The theory is associated especially with Karamata, who developed key results such as representation and integral theorems that clarify how slowly varying factors behave under integration and scaling. Subsequent work by multiple authors expanded the toolbox, including uniform convergence, bounds such as Potter’s, and inversion techniques involving conjugate functions.