By A. L. Yakimiv
Yakimiv (Steklov Institute of arithmetic) introduces Tauberian theorems and applies them to interpreting the asymptotic habit of stochastic techniques, checklist techniques, random diversifications, and infinitely divisible random variables. specifically, the ebook covers multidimensional extensions of Tauberian theorems as a result of Karamata, weakly oscillating capabilities, one-dimensional Tauberian theorems, Tauberian theorems because of Drozhzhinov and Zavyalov, Markov branching tactics, and chances of huge deviations within the context of the checklist version
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Extra info for Probabilistic Applications of Tauberian Theorems
X/ . tjxje/ . ;x/ e < dx ! C1; t ! 14). 9) is valid. 2. 2. For two cones 1 and 2 we write 1 2 if the closure of the set fxW x 2 1 ; jxj D 1g is contained in int 2 . The main result of this section is the following multidimensional Tauberian comparison theorem. 4. x/ deﬁned in , let there exist their Laplace transforms fy. / and gy. 4). If gy. t/=fy. t/ ! x/ ! 1; for any cone 0 t ! 0C ; jxj ! 17) . This Tauberian theorem extends those given in (Vladimirov, 1978; Stadtm¨uller, Trautner, 1979; Stadtm¨uller, Trautner, 1981; Stadtm¨uller, 1983).
Let > 0. u/ are true as t ! 1, and =t ! 0. The lemma is proved. 6. 1. t/ is weakly oscillating at inﬁnity. 4, as t ! 0; 1/. t/ A. t/ C B. t/ A. t/ B. t/ as t ! 3). 0; 1/ as t ! 4) t We take an arbitrary " > 0. 1 C "/1=3 1. 0; ı1 /. 6) for t t2 . t/ as t ! 0; ı2 / there exists t3 > t2 such that for any t3 b. 7) 54 1. 1 C ı/. 1 ı/t/ are true. The theorem is proved. 1, we derive the following Tauberian theorem. 2. 1 and let there exist M such that for any ﬁxed n n M n d y d f . 1// n gy. 10) t !
4). If gy. t/=fy. t/ ! x/ ! 1; for any cone 0 t ! 0C ; jxj ! 17) . This Tauberian theorem extends those given in (Vladimirov, 1978; Stadtm¨uller, Trautner, 1979; Stadtm¨uller, Trautner, 1981; Stadtm¨uller, 1983). P ROOF. We assume the contrary: let there exist a sequence xm 2 0 such that jxm j ! 1 as m ! xm / ! c; m ! 18) 44 1. Tauberian theorems We set xm D tm em , where tm D jxm j, em D xm =jxm j. Without loss of generality, we assume that em ! e 2 B D fxW x 2 ; jxj D 1g as m ! 1. We observe that e 2 G, because 0 .
Probabilistic Applications of Tauberian Theorems by A. L. Yakimiv