Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

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.