A generalization of Karamata’s theorem on the asymptotic behavior of integrals. (Ukrainian, English) Zbl 1224.26007

Buldygin, V. V.; Pavlenkov, V. V.

Teor. Jmovirn. Mat. Stat. 81, 13-24 (2009); translation in Theory Probab. Math. Stat. 81, 15-26 (2010).

J. Karamata in his articles [Mathematica 4, 38–53 (1930; JFM 56.0907.01); Bull. Soc. Math. Fr. 61, 55–62 (1933; Zbl 0008.00807)] introduced the notion of regularly varying (RV) functions and proved a number of fundamental theorems for them. The theorem on the asymptotic behaviour of integrals of RV functions is one of those results. This theorem found many applications in probability theory (see, for example, [L. de Haan, On regular variation and its application to the weak convergence of sample extremes. Amsterdam: Mathematisch Centrum (1970; Zbl 0226.60039); N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.26001)]).
Let \(A > 0\) and let \({\mathbb F}_+(A)\) be the set of positive and measurable functions \(f = (f(x), x\geq A)\). A function \(f\in{\mathbb F}_+(A)\) is called regularly varying (at infinity) in the Karamata sense if the limit \(\kappa_f (\lambda)=\lim_{x\to\infty}f(\lambda x)/f(x)\) exists and is positive and finite for all \(\lambda > 0\). We say that an RV function \(f\) is slowly varying (SV) if \(\kappa_f (\lambda)=1\) for all \(\lambda > 0\). If \(f\) is an RV function, then there is a real number \(\rho\) (called the index of the function \(f\)) such that \(\kappa_f (\lambda)=\lambda^{\rho}\), \(\lambda > 0\). Any RV function \(f\) of index \(\rho\) is represented in the form \(f(x) = x^{\rho}\ell(x), x\geq A\), where \(\ell(x)\) is a corresponding SV function. There are two parts of the Karamata theorem, namely a forward and a backward Karamata theorem.
Forward theorem. Let \(f\) be a locally integrable (integrable on every interval \([a, b]\subset [A,\infty)\)) regularly varying function of index \(\rho>-1\). Then \[ \int_A^{x}f(t)\,dt\underset{x\to\infty} \sim\frac{xf(x)}{\rho+1}. \] Backward theorem. Let \(f\in{\mathbb F}_+(A)\) be a locally integrable function. If there exists a number \(\gamma\in(0,\infty)\) such that \[ \int_A^{x}f(t)\,dt\underset{x\to\infty}\sim\frac{xf(x)}{\gamma}, \] then \(f\) is an RV function of index \(\rho=\gamma-1\).
Here, \(f(x)\underset{x\to\infty}\sim g(x)\) means that the functions \(f(x)\) and \(g(x)\) are asymptotically equivalent.
The authors of this paper generalized these Karamata theorems to the case of functions \(f\in{\mathbb F}_+(A)\) that have the form \(f(x) = x^{\rho}(x)\ell_0(x)H(\ln(x))\), \(x\geq A\), where \(\rho\in\mathbb R\), \(\ell=(\ell(x),\;x\geq A)\) is a slowly varying function and \(H=(H(u),\;u\in\mathbb R)\) is a positive continuous periodic function.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

26A12	Rate of growth of functions, orders of infinity, slowly varying functions
26A48	Monotonic functions, generalizations
34C41	Equivalence and asymptotic equivalence of ordinary differential equations

Keywords:

regularly varying function; Karamata theorem; asymptotic behavior of integrals; oscillating function

Citations:

Zbl 0226.60039; Zbl 0617.26001; JFM 56.0907.01; Zbl 0008.00807; JFM 59.0994.01