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Asymptotic behaviour of near-maxima of Gaussian sequences

Author

Listed:
  • Rasbagh Vasudeva
  • J. Vasantha Kumari

Abstract

Let $$(X_1,X_2,\ldots ,X_n)$$ ( X 1 , X 2 , … , X n ) be a Gaussian random vector with a common correlation coefficient $$\rho _n,\,0\le \rho _n>1$$ ρ n , 0 ≤ ρ n > 1 , and let $$M_n= \max (X_1,\ldots , X_n),\,n\ge 1$$ M n = max ( X 1 , … , X n ) , n ≥ 1 . For any given $$a>0$$ a > 0 , define $$T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)$$ T n ( a ) = j , 1 ≤ j ≤ n , X j ∈ ( M n - a , M n ] , K n ( a ) = # T n ( a ) and $$S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1$$ S n ( a ) = ∑ j ∈ T n ( a ) X j , n ≥ 1 . In this paper, we obtain the limit distributions of $$(K_n(a))$$ ( K n ( a ) ) and $$(S_n(a))$$ ( S n ( a ) ) , under the assumption that $$\rho _n\rightarrow \rho $$ ρ n → ρ as $$n\rightarrow \infty ,$$ n → ∞ , for some $$\rho \in [0,1)$$ ρ ∈ [ 0 , 1 ) . Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Rasbagh Vasudeva & J. Vasantha Kumari, 2014. "Asymptotic behaviour of near-maxima of Gaussian sequences," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(7), pages 861-866, October.
  • Handle: RePEc:spr:metrik:v:77:y:2014:i:7:p:861-866
    DOI: 10.1007/s00184-013-0468-2
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    References listed on IDEAS

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    1. Dembinska, Anna & Iliopoulos, George, 2012. "On the asymptotics of numbers of observations in random regions determined by order statistics," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 151-160, January.
    2. Hashorva, Enkelejd, 2003. "On the number of near-maximum insurance claim under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 37-49, February.
    3. Dembinska, Anna, 2010. "On numbers of observations near randomly indexed order statistics," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 309-317, March.
    4. Li, Y. & Pakes, Anthony G., 2001. "On the number of near-maximum insurance claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 309-323, June.
    5. Pakes, Anthony G. & Li, Yun, 1998. "Limit laws for the number of near maxima via the Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 395-401, November.
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