Limit theorems for products of sums of independent random variables

• Autorzy: Krajka T. K. , Rychlik Z.
• Języki publikacji: EN

Abstrakty

EN

Let {X_n; n ≥ 1} be a sequence of independent random variables with finite second moments and {N_n; n ≥ 1} be a sequence of positive integer-valued random variables. Write S_n. =Σⁿ_k-1(X_k−EX_k); n ≥1; and let N be a standard normal random variable. In the paper the convergences...[formula], are considered for some sequences {a_n} and {γ_n} of positive integer numbers such that S_n + a_n≥ 0 a.e. The case when γ_n are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3].

Słowa kluczowe

EN

lognormal distribution; randomly indexed product of sums; central limit theorem; self-normalized

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2010
• Tom: Vol. 30, Fasc. 1
• Strony: 73--85
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Krajka T. K.

• Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

autor

Rychlik Z.

• Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

Bibliografia

• [1] B. C. Arnold and J. A. Villaseñor, The asymptotic distribution of sums of records, Extremes 1 (1998), pp. 351-363.
• [2] X. W. Lu and Q. C. Qi, A note on asymptotic distribution of products of sums, Statist. Probab. Lett. 68 (2004), pp. 407-413.
• [3] T. X. Pang, Z. Y. Lin and K. S. Hwang, Asymptotics for self-normalized random products of sums of i.i.d. random variables, J. Math. Anal. Appl. 334 (2007), pp. 1246-1259.
• [4] V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Oxford Stud. Probab. 4, Clarendon Press, Oxford 1995.
• [5] V. V. Petrov, Sums of Independent Random Variables, Springer, Berlin 1975.
• [6] Q. C. Qi, Limit distributions for products of sums, Statist. Probab. Lett. 62 (2003), pp. 93-100.
• [7] G. Rempała and J. Wesołowski, Asymptotic for products of sums and U-statistics, Electron. Comm. Probab. 7 (2002), pp. 47-54.