Stochastic processes – Estimation of a random variable, independent decomposition

I have a question about an argument in this article: P.

Let $ S = (1, infty) times (-1,1) subset mathbb {R} ^ 2 $ and $ X = ( {X_t }, {P_x } _ {x in S}) $ to be a diffusion process on $ D $.

(Imagine something like a Brownian motion in $ (- infty, infty) times (-1,1) $ conditioned to hit $ {1 } times (-1,1) $)

We note by $ r (t) $, $ y (t) $ the first coordinate process of $ X $ and the second coordinate process of $ X $, respectively. Let $ tau_r = inf {t> 0 mid r_t = r } $ for $ r $ 1 $.

The author discusses whether
$$ text {(A)} quad lim_ {r to infty} inf_ {y in (-1,1)} P_ {r, y} left ( int_ {0} ^ { tau_1} r (s) ^ {- 2} , ds = infty right) = 1. $$

As proof, the following lemma seems to play a crucial role (See Proposition 1 in P).

Support. Let $ {Z_n } _ {n = 1} ^ { infty} $ be a sequence of non-negative bounded independent random variables. Then, $ sum_ {n = 1} ^ { infty} Z_n = infty $, $ Q $-as. Yes Yes $ sum_ {n = 1} ^ { infty} E ^ {Q}[Z_n]= infty $. Right here, $ E ^ {Q} $ denotes waiting w.r.t. $ Q $.

For simplicity, let's assume random variables $ { int _ { tau_k} ^ { tau_ {k-1}} 1 / r (s) ^ 2 , ds } _ {k = 2} ^ {n} $ are linked. Then, the random variables $ { int _ { tau_k} ^ { tau_ {k-1}} 1 / r (s) ^ 2 , ds } _ {k = 2} ^ {n} $ are non-negative, bounded and independent under $ P_ {n, y} ( cdot mid X _ { tau_2}, cdots X _ { tau_ {n-1}} $. Right here, $ P_ {n, y} ( cdot mid X _ { tau_2}, cdots X _ { tau_ {n-1}} $ denotes the conditional probability w.r.t. the random vector $ (X _ { tau_2}, cdots X _ { tau_ {n-1}}) $.

The random variable is broken down as follows

$$ int_ {0} ^ { tau_1} r (s) ^ {- 2} = sum_ {k = 2} ^ {n} int _ { tau_k} ^ { tau_ {k-1}} r (s) ^ {- 2} , ds, quadri P_ {n, y} ( cdot mid X _ { tau_2}, cdots X _ { tau_ {n-1}}) – text {as} $$

and considered as a sum of independent random variables.

The following condition is sufficient for (A) ?:

begin {align *}
text {(B)} quad lim_ {n to infty} inf_ {y in (-1,1)} E_ {n, y} left[sum_{k=2}^{n}int_{tau_k}^{tau_{k-1}}r(s)^{-2},ds mid X_{tau_2},cdots X_{tau_{n-1}} right]= infty.
end {align *}

Can we reach (A) if we combine (B) and Prop?