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Suppose random continuous real functions have unique zeros. If pointwise convergence in probability gives opposite strict limiting signs at the two ends of every sufficiently small interval about the target, the intermediate value theorem traps their zeros in those intervals with probability tending to one. The limiting function need not be continuous and the random functions need not be monotone.

If $X_n\stackrel{d}{\to }c$ for fixed real $c$, convergence of the distribution functions at $c-\epsilon$ and $c+\epsilon$ shows $\mathbb{P}(|X_n-c|>\epsilon )\to 0$. Thus convergence in distribution to a deterministic limit implies convergence in probability to that limit, even though the implication fails for nonconstant limits on a specified probability space.

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

Convergence in $L^2$ means $\mathbb{E}[(X_n-X{)}^2]\to 0$. It implies convergence in probability by Markov inequality.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.