  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, $\limsup_n (n\log\log
n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ 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 $C<\infty$ such that, both,
$E\min(h^2(X_1,X_2),u)<C\log\log u$ for all large u and
$sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty,
\|g\|_\infty<\infty\}< C$.
