# Stochastic processes – Estimation of a random variable, independent decomposition

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?