The LIL for canonical U-statistics of order 2
Abstract
Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm,
lim sup
n
(nloglogn
)
−1
|
∑
1<=i<j<=n
h(
X
i
,
X
j
)|<∞
a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists
C<∞
such that, both,
Emin(
h
2
(
X
1
,
X
2
),u)<Cloglogu
for all large u and
sup{Eh(
X
1
,
X
2
)f(
X
1
)g(
X
2
):|f(X)
|
2
<1,∥g(X)
∥
2
<1,∥f
∥
∞
<∞,∥g
∥
∞
<∞}<C
.