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XXXII International Seminar on Stability Problems for Stochastic Norwegian University of Institute of Informatics Faculty of Computational Science and Technology, Problems, Mathematics and Cybernetics, Trondheim Russian Academy of Sciences Moscow State University (NTNU) (IPI RAN) (CMC MSU) XXXII International Seminar on Stability Problems for Stochastic Models and VIII International Workshop \Applied Problems in Theory of Probabilities and Mathematical Statistics related to modeling of information systems" 16 { 21 June Trondheim, Norway Book of Abstracts Edited by Prof. Victor Yu. Korolev and Prof. Sergey Ya. Shorgin Moscow Institute of Informatics Problems, Russian Academy of Sciences 2014 The Organizing Committee of the XXXII International Seminar on Stability Problems for Stochastic Models V. Zolotarev (Russia) – Honorary Chairman, V. Korolev (Russia) – Chairman, N. Ushakov (Norway) – Deputy Chairman, I. Shevtsova (Russia) – General Secretary, Yu. Khokhlov (Russia), S. Shorgin (Russia), S. Baran (Hungary), V. Bening (Russia), A. Bulinski (Russia), J. Misiewich (Poland), E. Omey (Belgium), G. Pap (Hungary), A. Zeifman (Russia), Yu. Nefedova (Russia) – Secretary. The Organizing Committee of the VIII International Workshop “Applied Problems in Theory of Probabilities and Mathematical Statistics related to modeling of information systems” I. Atencia (Spain), A. Grusho (Russia), K. Samouylov (Russia), S. Shorgin (Russia), S. Frenkel (Russia), R. Manzo (Italy), A. Pechinkin (Russia), N. Ushakov (Norway), E.Timonina (Russia) et al. XXXII International Seminar on Stability Problems for Stochastic Models (ISSPSM’2014) and VIII International Workshop “Applied Problems in Theory of Probabilities and Mathematical Statistics related to modeling of information systems” (APTP + MS’2014). Book of abstracts. – M.: IPI RAN, 2014. - 147 p. The book includes the abstracts of communications submitted to the XXXII International Seminar on Stability Problems for Stochastic Models (ISSPSM’2014) and associated VIII International Workshop “Applied Problems in Theory of Probabilities and Mathematical Statistics related to modeling of information systems” (APTP + MS’2014) (Summer Session). XXXII Международный семинар по проблемам устойчивости сто- хастических моделей (ISSPSM’2014) и VIII Международный рабочий семинар “Прикладные задачи теории вероятностей и математической статистики, связанные с моделированием информационных систем” (APTP + MS’2014). Сборник тезисов. – M.: ИПИ РАН, 2014. - 147 с. В сборник включены тезисы докладов, представленных на XXXII Меж- дународный семинар по проблемам устойчивости стохастических моделей (ISSPSM’2014) и VI Международный рабочий семинар “Прикладные задачи тео- рии вероятностей и математической статистики, связанные с моделированием информационных систем” (APTP + MS’2014) (летняя сессия). c Institute of Informatics Problems, Russian Academy of Sciences, 2014 c Authors, 2014 XXXII International Seminar on Stability Problems for Stochastic Models On special empirical processes of independence Abdurahim Abdushukurov 1, Leyla Kakadjanova 2 1National University of Uzbekistan, Uzbekistan, a [email protected] 2National University of Uzbekistan, Uzbekistan, leyla [email protected] Consider a following model of experiments in which observed pairs are consists of f(Xk;Ak) ; k > 1g, where Xk are random elements defined on a probability space (Ω; A; P) with values in a measurable space (X; B). Events Ak have a common probability p 2 (0; 1). Let δk = I (Ak) is indicator of the event Ak. At the n− th step of experiments is observed a sample (n) (n) S = f(Xk; δk) ; 1 6 k 6 ng. Each pair in the sample S induced a statisti- cal model with sample space X⊗f0; 1g, σ-algebra of sets of the form B×D and ∗ induced distribution Q (B × D) = P (Xk 2 B; δk 2 D), where B 2 B, D ⊂ ∗ ∗ f0; 1g. Define submeasures Q1 (B) = Q (B × f1g), Q0 (B) = Q (B × f0g) ∗ and Q (B) = Q (B × f0; 1g) = Q0 (B) + Q1 (B), B 2 B and its esti- n n 1 P 1 P mates Q1n (B) = n δkI (Xk 2 B) ; Q0n (B) = n (1 − δk)I (Xk 2 B) ; k=1 k=1 n 1 P Qn (B) = n I (Xk 2 B) = Q0n (B) + Q1n (B) for all B 2 B. Consider the k=1 hypothesis H of independence of Xk and Ak for each k > 1 . We consider general classes of specially normalized empirical processes of independence indexed by the class F of measurable functions f : X ! R. For a signed measure G and function f 2 F denote the integral Z Gf = f dG: X Define F - indexed empirical process Gn : F 2 R as: n p 1 − =2 X f 7! Gnf = n (Qn − Q) f = n (f (Xk) − Qf); f 2 F: k=1 p Note that Gnf = G0nf + G1nf, where Gjnf = n (Qjn − Qj ) f; j = 0; 1. Donsker-type theorems provide a general conditions on F , under which 1 Gnf ) Gf in l (F); (1) 1 where l (F) - the space of all bounded functions f : X ! R equipped with the supremum - norm kfkF and ) means the weak convergence. Limiting field fGf; f 2 Fg called Q - Brownian bridge. In connection with the problem of testing the hypothesis H, we introduce F - processes Λf = Q1f −pQf; Λnf = = Q1nf − pnQnf; f 2 F and Z n 1=2 ∆nf = fd∆n = (Λn − Λ) f; f 2 F: (2) pn (1 − pn) X 3 XXXII International Seminar on Stability Problems for Stochastic Models One of important properties of the process (2) is its convergence to the same Q-Brownian bridge fGf ; f 2 Fg under validity of H . To present the basic theorems we define the complexity or entropy of class F. Bracketing (or covering) number N[] ("; F; Lq (Q)) is the minimum number of " - brackets in Lq(Q) needed to cover F (see Shorack and Wellner [1], Van der Vaart and Wellner [2]): ( k : for some f1; :::; fk 2 Lq (Q) ; N ("; F; Lq ( )) = min [] Q F ⊂ [ [fi; fj ]: kfj − fik ": i;j Q;q 6 For weak convergence of F - indexed empirical processes (2) we need the integral of the metric entropy with bracketing to be δ Z (q) 1=2 Jj[] (δ) = Jj[] (δ; F; Lq (Qj )) = (Hjq (")) d"; j = 0; 1; for 0 < δ < 1; 0 where Hjq (") = log N[] ("; F; Lq (Qj )) is metric entropy of class F in Lq(Qj ); j = 0; 1. We introduce the following conditions: (i) Let the class F such that (2) F ⊂ L2(Qj ) and Jj[] (1) < 1; j = 0; 1: (3) Theorem 1. Under the conditions (3) for n ! 1 1 ∆nf ) ∆f in l (F); (4) where f∆f; f 2 Fg is a Gaussian field with zero mean and under validity of the hypothesis H, it coincides in distribution with Q - Brownian bridge. Now concider case of random sample size. Let the sequence Nn of Poisson r.v.-s with mean n. Suppose that sequences fNn; n ≥ 1g and ∗ f(Xk; δk); k ≥ 1g are independent. Let ∆Nn f; f 2 Fg be sequence of normalized empirical processes of independence obtained from (2) by replacing upper index n of all summation to a random sequence Nn . Theorem 2. Under the conditions (3) at n ! 1 ∗ ∗ 1 ∆Nn f ) ∆ f in l (F); (5) ∗ d where by hypothesis H, ∆ f = W(f), f 2 F. Here fW(f); f 2 Fg is Brownian sheet. References 1. G. R. Shorack, J. A. Wellner. Empirical processes with applications to statistics, 1986, John Wiley&Sons. 4 XXXII International Seminar on Stability Problems for Stochastic Models 2. A. W. Van der Vaart, J. A. Wellner Weak convergence and empirical pro- cesses, 1996, Springer. High level subcritical branching processes in a random environment Valeriy Afanasyev 1 1Steklov Mathematical Institute, Russia, [email protected] Let fξn; n 2 N0g be a branching process in a random environment (BPRE) defined by a sequence of independent and identically distributed (random) gen- erating functions ffn (s) ; n 2 Ng. Note that ξn is the size of the nth generation (we assume that ξ0 = 1). The generating function fn (s), s 2 [0; 1], defines the reproduction law for the particles in the (n − 1)th generation, n 2 N. 0 0 Assuming that f1 (1) 2 (0; +1) a.s., we set Xi = ln fi (1) for i 2 N. Note that the random variables X1;X2;::: are independent and identically Pn distributed. Introduce the associated random walk S0 = 0, Sn = i=1 Xi, n 2 N. Suppose that the process fξng is subcritical, i.e. EX1 < 0, and there exists a positive number { such that E exp ({X1) = 1; E (jX1j exp ({X1)) < +1: (1) Condition (1) is classical for random walks with negative drift and allows one to pass to conjugate random walk with positive drift. In addition, we assume that E (ξ1 ln (ξ1 + 1) exp (({ − 1) X1)) < +1; (2) and if { ≥ 1, then there exists a number p > { such that p E (ξ1 exp (({ − p) X1)) < +1: (3) Introduce the first passage time of the process fξng to a level x > 1: Tx = min fn : ξn > xg ; and the lifetime of the process fξng: T = min fn : ξn = 0g . In [1] and [2], the author showed that if conditions (1)-(3) are satisfied, then −{ P (Tx < +1) ∼ c0x ; Tx P 1 Tx < +1 ! ; ln x a 5 XXXII International Seminar on Stability Problems for Stochastic Models T P 1 1 Tx < +1 ! − ln x a b as x ! +1, where c0 is a positive constant, a = E (X1 exp ({X1)), b = EX1. In addition, we assume that 2 E X1 exp ({X1) < +1: (4) 2 2 2 Set σ = E X1 exp ({X1) − a . Let B = fB (t) ; t 2 [0; 1]g be a standard Brownian motion. The following functional limit theorem for the first passage time to different levels is valid. Theorem 1. If fξn; n 2 N0g is a subcritical BPRE and conditions (1)-(4) hold, then Txt − t ln x=a D p ; t 2 [0; 1] Tx < +1 ! B σa−3=2 ln x as x ! +1, where the symbol !D means convergence in distribution in the space D [0; 1] with the Skorokhod topology.
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