Slowly varying function

In real analysis, a branch of mathematics, a slowly varying function is a function resembling a function converging at infinity. Slowly varying functions are important in probability theory.

Definition

A function $L\; colon\; (0,infty)\; o\; (0,infty)$ is called "slowly varying" (at infinity) if for all "a" > 0,:$lim\_\{x\; o\; infty\}\; frac\{L(ax)\}\{L(x)\}=1.$If this limit is finite but nonzero for every "a" > 0, then the function "L" is a regularly varying function. These definitions are due to Jovan Karamata harv|Galambos|Seneta|1973.

Examples

* If $lim\_\{x\; o\; infty\}\; L(x)\; =\; b\; in\; (0,infty),$ then $L$ is a slowly varying function.
* For any , is slowly varying.
* The function $L(x)=x$ is not slowly varying, neither is for any real

Properties

Some important properties are harv|Galambos|Seneta|1973:
* The limit in the definition is uniform if "a" is restricted to a finite interval.
* Every regularly varying function is of the form "x" ^"β""L"("x") where "β" ≥ 0 and "L" is a slowly varying function.
* For every slowly varying function "L", there exists "B" > 0 such that for all "x" ≥ "B" the function can be written in the form

::$L(x)\; =\; exp\; left(\; eta(x)\; +\; int\_B^x\; frac\{varepsilon(t)\}\{t\}\; ,dt\; ight)$

:where "η"("x") converges to a finite number and "ε"("x") converges to zero as "x" goes to infinity.

References

*.