Self-Normalized Moderate Deviations for Degenerate U -Statistics.

Entropy. Jan2025, Vol. 27 Issue 1, p41. 27p.

In this paper, we study self-normalized moderate deviations for degenerate U-statistics of order 2. Let { X i , i ≥ 1 } be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form h (x , y) = ∑ l = 1 ∞ λ l g l (x) g l (y) , where λ l > 0 , E g l (X 1) = 0 , and g l (X 1) is in the domain of attraction of a normal law for all l ≥ 1 . Under the condition ∑ l = 1 ∞ λ l < ∞ and some truncated conditions for { g l (X 1) : l ≥ 1 } , we show that log P ( ∑ 1 ≤ i ≠ j ≤ n h (X i , X j) max 1 ≤ l < ∞ λ l V n , l 2 ≥ x n 2) ∼ − x n 2 2 for x n → ∞ and x n = o (n) , where V n , l 2 = ∑ i = 1 n g l 2 (X i) . As application, a law of the iterated logarithm is also obtained. [ABSTRACT FROM AUTHOR]

*Random variables, *Kernel functions, *U-statistics, *Logarithms

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