some $L\ge \ell\ge 0$ and $q\in(0,1)$. Then \[eq:lem.SubH.IID\] f(u) q\^[ ]{} (f, ) q\^ (f, ), (f, ) := \_[x]{} |f(x)|.

\[lem:BiSubH.IID\] Let $ \ball\subset\DZ^d$ a finite connected subgraph, $\ball_{r'}(u')$, $\ball_{r''}(u'') \subset\ball$ two disjoint balls with $|u'-u''|\ge r' + r'' +2$, and $f:\ball\times\ball\to \DR_+$, $(x',x'')\mapsto f(x',x'')$ a function which is separately $(\ell,q)$-subharmonic in $x'\in\ball_{r'}(u')$ and in $x''\in\ball_{r''}(u'')$. Then \[eq:RDL.IID\] f(u’, u”) q\^[ + ]{} (f, ) q\^[ ]{} (f, ).

\[def:S.IID\] Given a sample $V(\cdot;\om)$, a ball $ \ball_L(u)$ is called $(E,m)$-non-singular ($(E,m)$-NS) if for any pair of points $x,y\in \ball_L(u)$ with $|x-y|\ge L^{\frac{1+\rho}{\alpha}} = L^{7/8}$, \[eq:def.NS.IID\] |\_L(u)| |G(x,y;E;)| e\^[-(m,L) |x - y| + 2L\^]{}, where \[eq:def.gamma.IID\] (m,L) := m(1 + L\^[-1/8]{}). Otherwise, it is called $(E,m)$-singular ($(E,m)$-S).

Observe that, with $m\ge 1$ and $L$ large enough, \[eq:gamma.7.8\] (m,L) |x - y| - 2L\^ &= |x - y| m ( 1 + L\^[-]{} - )\
& |x - y| m ( 1 + L\^[-]{} - 2 L\^[- 1 -]{} )\
&m ( 1 + L\^[-]{} )|x - y|, provided that $1+\rho - \beta > \tau$, which is the case with $\rho = 1/6$, $\beta=1/2$ and $\tau=1/8$. Therefore, the $(E,m)$-NS property implies that \[eq:def.NS.IID.implies\] |\_L(u)| |G(u,y;E;)| e\^[-m (1 + L\^[-]{}) |x 