Erratum to: The Asymptotic Distribution of Self-Normalized Triangular Arrays
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Erratum to: The Asymptotic Distribution of Self-Normalized Triangular Arrays David M. Mason
Published online: 9 March 2013 © Springer Science+Business Media New York 2013
Abstract We correct and clarify some ambiguous statements in D. M. Mason (2005): The asymptotic distribution of self-normalized triangular arrays. J. Theoret. Probab., 18, 853–870. Corrections and Clarifications of Mason (2005) This note has two purposes. First is to correct some statements in the Introduction and Statements of Results of [4], and second is to provide the result given in Proposition [A] below, which clarifies a claim at the end of the proof of Theorem 2. Our corrections are needed since it is not clear that (1.9) always implies (1.2). They are the following: (i) On page 855, line 10, change the “further shows” to “further shows that under the setup of Proposition [A] in this note” and on line 13 change “(1.4)” to “(1.4) nondegenerate”. (ii) On page 855, line 14, replace “or equivalently (1.9) holds” with “P (V > 0) = 1”. (iii) On page 856, line 3, replace “ Actually” with “Actually under the setup of Proposition [A] in this note”. We remark in passing that the statements in [4] about triangular arrays of the form X 1,n , . . . , X n,n , n ≥ 1, are equally valid for triangular arrays of the form X 1,n k , . . . , X n k ,n k , k ≥ 1, where {n k }k≥1 is an infinite subsequence of the positive integers. Also we point out that everywhere triangular array of infinitesimal independent
The online version of the original article can be found under doi:10.1007/s10959-005-7529-z. D. M. Mason (B) Department of Applied Economics and Statistics, University of Delaware, Newark, DE 19716, USA e-mail: [email protected]
123
590
J Theor Probab (2013) 26:589–593
random variables should be changed to infinitesimal triangular array of independent random variables, as stated in Proposition [A]. The word infinitesimal was everywhere put in the wrong place. The following Proposition [A] and Remark 1 justify the claim towards the end of the proof of Theorem 2 on page 868 that says, “This means that every subsequential distributional limit random variable T must be of the form (1.10).” They should have been included in an appendix in the original paper. Proposition [A] Let {n k }k≥1 be an infinite subsequence of the positive integers and X 1,n k , . . . , X n k ,n k , k ≥ 1, be an infinitesimal triangular array of independent random variables such that for each k ≥ 1, X 1,n k , . . . , X n k ,n k are i.i.d. X 1,n k . Assume that for a necessarily infinitely divisible random variable U , nk
X i,n k →d U, as k → ∞.
(1.1)
i=1
Then n k
X i,n k ,
i=1
nk
2 X i,n k
→d (U, V ) , as k → ∞,
(1.2)
i=1
where the two dimensional infinitely divisible random vector (U, V ) in (1.2) has the representation: (U, V ) =d b + W + τ Z , S + τ 2 ,
(1.3)
with b and τ ≥ 0 being suitable constants,
1
W = 0
ϕ1 (s) d N1 (s) +
∞
1
1
−
ϕ1 (s) d {N1 (s) − s}
ϕ2 (s) d N2 (s) −
0
∞
ϕ2 (s) d {N2 (s) − s}
(1.4)
ϕ22 (s) d N2 (s),
(1.5)
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