Self-Normalized Moderate Deviations for Degenerate U-Statistics - Lin Ge; Hailin Sang; Qi-Man Shao (Entropy)
In this paper, we study self-normalized moderate deviations for degenerate U-statistics of order 2. Let <semantics>{Xi,i≥1}</semantics> be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form <semantics>h(x,y)=<msubsup>∑l=1∞</msubsup>λlgl(x)gl(y)</semantics>, where <semantics>λl>0</semantics>, <semantics>Egl(X1)=0</semantics>, and <semantics>gl(X1)</semantics> is in the domain of attraction of a normal law for all <semantics>l≥1</semantics>. Under the condition <semantics><msubsup>∑l=1∞</msubsup>λl<∞</semantics> and some truncated conditions for <semantics>{gl(X1):l≥1}</semantics>, we show that <semantics>logP(<mstyle scriptlevel="0" displaystyle="true">∑1≤i≠j≤nh(Xi,Xj)max1≤l<∞λl<msubsup>Vn,l2</msubsup></mstyle>≥<msubsup>xn2</msubsup>)∼−<mstyle scriptlevel="0" displaystyle="true"><msubsup>xn2</msubsup>2</mstyle></semantics> for <semantics>xn→∞</semantics> and <semantics>xn=o(<msqrt>n</msqrt>)</semantics>, where <semantics><msubsup>Vn,l2</msubsup>=<msubsup>∑i=1n</msubsup><msubsup>gl2</msubsup>(Xi)</semantics>. As application, a law of the iterated logarithm is also obtained.