Improvement of the Basic Lemma and resultsĮverywhere in this section we suppose the condition (4) holds. Subsequently of this we will improve the Lemma 1.2 and define a speed rate in some well-known limit theorems from the theory of critical GW branching process.Ģ. We devote this paper to improvement of the Lemma 1.1 provided that the condition (4) holds with given a(x). In this case L(x) is called SV with remainder, see. Henceforth we suppose that some positive function g(x) is given so that g(x) ^ 0 and a(x) = o(g(x)) as x ^ x>. Watson (GW) branching process, where N0 = G(x), where G(x) has the Laplace transformīy arguments of Slack one can be shown that if the condition (1) holds thenįor 0 0, where a(x) ^ 0 as x ^ x>. Let F(s) = Pjsj denote an offspring probability generating function (PGF) of Galton. We refer the reader to and for more information. A function V(x) is said to be regularly varying at infinity with index of regular variation p G R+ if it in the form V(x) = xpl(x), where l(x) is SV at infinity. Remind that real-valued, positive and measurable function l(x) is said to be SV at infinity in sense of Karamata if £(Xx)/£(x) ^ 1 as x ^ x> for each A > 0. Afterwards Slack and Seneta, prove principally new limit theorems for branching processes using slowly varying (SV) functions. Zolotarev one of the first demonstrated an encouraging perspective of application of the conception of slow variation in probability theory, in particular in the theory of stochastic branching processes.

DOI: 10.17516/1997-.Ī conception of slow variation (or more general - regular variation) was initiated first by Jovan Karamata in. Keywords: Galton-Watson branching process, slowly varying functions, generating functions. We improve the Basic Lemma of the theory of critical Galton-Watson branching processes and refine some well-known limit results. Consider the critical case so that the generating function of the per-capita offspring distribution has the infinite second moment, but its tail is regularly varying with remainder. Received, received in revised form, accepted We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Galton-Watson branching processes. Karshi State University 17, Kuchabag st., Karshi city, 180100 Uzbekistan State Testing Center under the Cabinet of Ministers of the Republic of Uzbekistanġ2, Bogishamol st., 100202, Tashkent Karshi State University 17, Kuchabag st., Karshi city, 180100 Uzbekistan On Application of Slowly Varying Functions with Remainder in the Theory of Galton-Watson Branching Process

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