ISSN:
1572-9230
Keywords:
Banach space
;
compact law of the iterated logarithm
;
independent random variables
;
two-dimensional indices
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, X n, m; (n, m) ∈ N 2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$ . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N 2 is studied. There is a gap between the moment conditions for CLIL(N 1) and those for CLIL(N 2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, N r (α, φ) = {(n, m) ∈ N 2; n α ≤ m ≤ n α exp{(log n) r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ 〉 0.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022687923455
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