On stability of trimmed sums

Hu, T.-C.; Huang, C.-Y.; Rosalsky, A.

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Tytuł artykułu

On stability of trimmed sums

Autorzy

Hu T.-C. , Huang C.-Y. , Rosalsky A.

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Abstrakty

Let {X_n, n ≥ 1} be a sequence of i.i.d. random variables and let {a_n, n ≥ 1} and {b_n, n ≥ 1} be sequences of constants where 0 < b_n ↑ ∞. Let X_n⁽¹⁾, X_n⁽²⁾,…, X_n⁽ⁿ⁾ be a rearrangement of X₁,…, X_n such that |X_n⁽¹⁾| ≥ |X_n⁽²⁾| ≥ … ≥ |X_n⁽ⁿ⁾|. Consider the sequence of weighted sums T_n = Σⁿ_i=1 a_i X_i, n ≥ 1, and, for fixed r ≥ 1, set T^(r)_n = Σⁿ_i=1 a_i X_i I(|X_i| ≤ |X^(r+1)_n|), n ≥ r + 1; i.e., T^(r)_n is the sum T_n minus the sum of the X^(k)_n’s multiplied by their corresponding coefficients for k = 1,…, r. The main results provide sufficient and, separately, necessary conditions for b⁻¹_n T^(r)_n − k_n → 0 almost surely for some sequence of centering constans {k_n, n ≥1}. The current work extends that of Mori [14], [15] wherein a_n ≡ 1.

Słowa kluczowe

extreme terms lightly trimmed sums almost sure convergence strong law of large numbers

Wydawca

Wrocław University of Science and Technology

Czasopismo

Probability and Mathematical Statistics

Rocznik

2003

Tom

Vol. 23, Fasc. 1

Strony

153--172

Opis fizyczny

Bibliogr. 20 poz.

Twórcy

autor

Hu T.-C.

tchu@math.nthu.edu.tw

Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 300, R.O.C.

autor

Huang C.-Y.

Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 300, R.O.C.

autor

Rosalsky A.

rosalsky@stat.ufl.edu

Department of Statistics, University of Florida, Gainesville, Florida 32611, U.S.A.

Bibliografia

[1] A. Adler and A. Rosalsky, On the strong law of large numbers for normed weighted sums of i.i.d. random variables, Stochastic Anal. Appl. 5 (1987), pp. 467-483.
[2] V. Barnett and T. Lewis, Outliers in Statistical Data, 2nd edition, Wiley, Chichester, Great Britain, 1984.
[3] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd edition, Springer, New York 1997.
[4] S. Csörgő and G. Simons, A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games, Statist. Probab. Lett. 26 (1996), pp. 65-73.
[5] J. H. J. Einmahl and E. Haeusler, On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics, Probab. Theory Related Fields 79 (1988), pp. 59-74.
[6] W. Feller, An extension of the law of the iterated logarithm to variables without variance, J. Math. Mech. 18 (1968), pp. 343-355.
[7] H. Kesten, Convergences in distribution of lightly trimmed and untrimmed sums are equivalent, Math. Proc. Cambridge Philos. Soc. 113 (1993), pp. 615-638.
[8] H. Kesten and R. A. Maller, Ratios of trimmed sums and order statistics, Ann. Probab. 20 (1992), pp. 1805-1842.
[9] H. Kesten and R. A. Maller, Infinite limits and infinite limit points of random walks and trimmed sums, Ann. Probab. 22 (1994), pp. 1473-1513.
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[11] P. Lévy, Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînée, J. Math. Pures Appl. 14 (1935), pp. 347-402.
[12] R. A. Maller, Asymptotic normality of lightly trimmed means - a converse, Math. Proc. Cambridge Philos. Soc. 92 (1982), pp. 535-545.
[13] R. A. Maller, Relative stability of trimmed sums, Z. Wahrsch. Verw. Gebiete 66 (1984), pp. 61-80.
[14] T. Mori, The strong law of large numbers when extreme terms are excluded from sums, Z. Wahrsch. Verw. Gebiete 36 (1976), pp. 189-194.
[15] T. Mori, Stability for sums of i.i.d. random variables when extreme terms are excluded, Z. Wahrsch. Verw. Gebiete 40 (1977), pp. 159-167.
[16] T. Mori, On the limit distributions of lightly trimmed sums, Math. Proc. Cambridge Philos. Soc. 96 (1984), pp. 507-516.
[17] S. V. Nagaev, On necessary and sufficient conditions for the strong law of large numbers, Teor. Verojatnost. i Primenen. 17 (1972), pp. 609-618 (in Russian). English translation in: Theory Probab. Appl. 17 (1972), pp. 573-581.
[18] V. I. Pozdnyakov, On the strong law of large numbers lor reduced sums, Vestnik S.-Peterburg. Univ. Mat. Mekh. Astronom., vyp. 2 (1994), pp. 20-25 (in Russian). English translation in: Vestnik St. Petersburg Univ. Math. 27, No. 2 (1994), pp. 16-20.
[19] Yu. V. Prokhorov, An extremal problem in probability theory, Teor. Verojatnost. i Primenen. 4 (1959), pp. 211-214 (in Russian). English translation in: Theory Probab. Appl. 4 (1959), pp. 201-203.
[20] W. F. Stout, Almost Sure Convergence, Academic Press, New York 1974.

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Bibliografia

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