Karamata’s Integral Representation

• Karamata’s integral representation is a framework that generalizes slowly varying functions by providing two-sided integral representations for a broader class of measurable functions.
• It uses a non-decreasing, admissible scale function ψ, satisfying ψ(x)=o(x), to unify various function classes including long-tailed and slowly varying functions.
• The representation facilitates precise asymptotic analysis of tail probabilities and large deviation behavior via the extended Borovkov–Borovkov theorem.

Karamata’s integral representation generalizes the classic analytic description of slowly varying functions by providing two-sided integral representations for a broader class of measurable functions, including those that are neither strictly slowly varying nor long-tailed. The extension, defined for $\psi$-locally constant functions, enables precise characterizations of tails of distributions and the analysis of large deviation behaviors in probability theory. The fundamental representation theorem and its corollaries, attributable to Borovkov and Borovkov, strictly subsume classical Karamata results by accommodating an intermediate scale function $\psi(x)=o(x)$ and prescribing structural properties for this scale.

1. Definitions and Classes of Functions

A function $\psi:[0,\infty)\rightarrow[1,\infty)$, non-decreasing and $\psi(x)=o(x)$ as $x\to\infty$, is called an admissible scale function. Given such a $\psi$, a positive measurable function $f:[0,\infty)\rightarrow(0,\infty)$ is defined as $\psi$–locally constant (abbreviated $\psi$–l.c.f.) if, for any fixed $v\in\mathbb{R}$, the following limit holds whenever $x + v\psi(x) \geq \lambda x$ for some $\lambda > 0$ for all sufficiently large $x$:

$\lim_{x\to\infty} \frac{f(x+v\psi(x))}{f(x)} = 1.$

Notably, $\psi(x) \equiv 1$ recovers the concept of long-tailed functions while $\psi(x) = x$ is the classical slowly varying case. The class $K$ of admissible $\psi$ is comprised of functions that satisfy two additional conditions: (A) For each $v>0$, there exists $a_0=a(v)>0$ and $x_0<\infty$ such that for all $x \geq x_0$,

$\psi(x - v\psi(x)) \geq a(v)\psi(x),$

and (B) $\int_0^{\infty} a(u) du = \infty$.

2. Integral Representation Theorem for $\psi$–Locally Constant Functions

The central representation due to Borovkov–Borovkov states: for $\psi\in K$, a positive measurable function $f$ is $\psi$–locally constant if and only if there exist measurable functions $c(x)>0$ and $\epsilon(x)$ with $c(x)\to c\in(0,\infty)$ and $\epsilon(x)\to 0$ as $x\to\infty$, such that for $x\geq 1$,

$$f(x) = c(x)\exp\left\{\int_1^x \frac{\epsilon(t)}{\psi(t)}\,dt\right\}.$$

This result strictly extends both the Karamata representation for slowly varying functions and the integral formula for locally constant (long-tailed) functions, by encompassing the entire intermediate regime described by arbitrary $\psi\in K$ (Borovkov et al., 2010).

As an immediate corollary, any $\psi$–locally constant $f$ satisfies

$$f(x) = \exp\{ o(\int_1^x dt/\psi(t))\}\quad \text{as }x\to\infty,$$

implying sub-exponential growth relative to the scale set by $\psi$.

3. Connections to Classical Theory

Specializing $\psi(x)$ recovers classical results:

• For $\psi(x) = x$ (slowly varying functions), the representation becomes:

$$L(x) = c(x)\exp\left\{\int_1^x \frac{\epsilon(t)}{t}dt\right\},$$

which is the classic Karamata form.

• For $\psi(x) = 1$ (long-tailed or locally constant functions):

$$g(x) = c(x)\exp\left\{\int_1^x \epsilon(t)dt\right\},$$

matching the integral form of the l.c.f. representation.

The Borovkov–Borovkov theorem thus strictly generalizes both, allowing arbitrary non-decreasing $\psi=o(x)$, including regularly varying scales $\psi(x)=x^{\alpha}L(x)$ for $\alpha<1$. This strict extension is justified by the structure of the representation, which reduces to either limiting case for the respective choices of $\psi$.

4. Outline of Proof

The proof strategy is as follows:

• Uniformity of the $\psi$–local constancy ratio: The convergence in the definition of $\psi$–l.c.f. is shown to be uniform in $v$ on any compact interval by a Delange-type covering argument, employing hypothesis (A) to handle negative $v$.
• Reduction to the l.c.f. case: Introducing $y(x) = \int_1^x dt/\psi(t)$ establishes a change of variable in which the property of $\psi$-local constancy is transformed to classic local constancy for $F(t) = f(y^{-1}(t))$. The classical integral representation for l.c.f.'s is then invoked to yield the desired form.
• Converse: If $f$ is of the integral form stated above, then direct estimation shows that the ratio $f(x+v\psi(x))/f(x)\to 1$ as $x\to\infty$ for all fixed $v$.

For scales $\psi$ that are regularly varying with index $\alpha<1$ and satisfy additional smoothness conditions ($\psi\in K_1$), the variable $y(x)$ can be replaced by

$\Theta(x) = \int_1^x dt/\psi(t),$

yielding a structurally parallel, but sometimes more transparent, representation:

$$f(x) = c(x)\exp\left\{\int_1^x \tilde{\epsilon}(t) d\Theta(t)\right\},\;\; \tilde{\epsilon}(t)\to 0.$$

5. Examples, Special Cases, and Corollaries

The integral representation framework yields the following immediate special cases and corollaries:

$\psi(x)$	Resulting Representation for $f(x)$	Interpretation
$1$	$c(x)\exp\{\int_1^x \epsilon(t)dt\}$	Locally constant (l.c.f.)
$x$	$c(x)\exp\{\int_1^x \epsilon(t)/t\,dt\}$	Slowly varying (Karamata)
$x^\alpha L(x), \alpha<1$	$c(x)\exp\{\int_1^x \tilde{\epsilon}(t)d\Theta(t)\}$	Regularly varying scales

Any $\psi$–locally constant $f$ satisfies $f(x) = \exp\{ o(\int_1^x dt/\psi(t)) \}$. This condition constrains the growth of $f$ to lie below any exponential of the characteristic integral of $1/\psi$.

6. Implications for Large Deviation Theory

The generalization of Karamata’s integral representation is critical in large deviation theory, especially for distributions whose tails are not regularly varying but merely $\psi$–locally constant. In particular, the extension allows classical one-term large deviation asymptotics to apply to sums of random variables with distribution tails in this larger class. Theorems B and C in Borovkov–Borovkov (Borovkov et al., 2010) demonstrate that tail probabilities $P(S_n > x)$ can be analyzed with nearly the same precision as in the regularly varying setting, provided the tail functions exhibit $\psi$–local constancy.

7. Comparative Summary and Significance

The integral representation for $\psi$–locally constant functions provides a unified analytic tool that covers the entire spectrum between long-tailed and slowly varying functions, depending on the admissible scale $\psi$. The structure,

$$f(x) = c(x)\exp\left\{\int_1^x \frac{\epsilon(t)}{\psi(t)}\,dt\right\},$$

enables precise fine-tuning of asymptotic behaviors and has explicit utility in advanced probability, analytic number theory, and statistical applications where intermediate scaling between slow variation and constancy is encountered. This broad perspective facilitates extensions of classical large deviation theorems and further demonstrates the flexibility and depth of Karamata-type analytic approaches (Borovkov et al., 2010).