Quantile optimization for heavy-tailed distributions ... - CASTLE Lab

6 QUANTILE OPTIMIZATION FOR HEAVY-TAILED DISTRIBUTIONS
Since the left-hand side is finite and
we must have
Since β0,n ≥ 0 and
we conclude that
N
n=1
(γn−1) 2 λn +
∞
m=N
(γm) 2 ≤
∞
(γm) 2 < ∞,
n=1
∞
γn−1β0,n < ∞.
n=1
∞
γn−1 > ∞,
n=1
lim inf
n→∞ β0,n = lim
n→∞ E [βn−1,n | F0] = 0.
Therefore there is a subsequence (n∗ ) ⊆ (n) ∞
n=0 such that
lim
n∗ β0,n∗ = lim
→∞ n∗→∞ E [βn∗−1,n∗ | F0] = 0.
However, since βn ∗ −1,n∗ is a non-negative random variable, this implies that βn ∗ −1,n∗
goes to 0 with probability 1 as n ∗ → ∞. In other words,
Since
lim
n∗→∞ βn∗−1,n∗ = lim
n→∞ (Yn∗−1 − qα) E [sgnα (Yn∗−1 − Xn∗) | Fn∗−1] a.s.
= 0.
if and only if Yn ∗ −1 = qα,
E [sgn α (Yn ∗ −1 − Xn ∗) | Fn ∗ −1] = 0
lim
n∗ a.s.
Yn∗ = qα. (2.6)
→∞
This proves our goal but only for the subsequence (Yn∗). In order to get the convergence
of the whole sequence (Yn) ∞
n=0 we have to look back at equation (2.4) from
which we obtain that
where
lim
n→∞ |Yn − qα| a.s.
= W∞, (2.7)
0 ≤ W∞ < ∞.
Let Y + n , Y − n be the the random variables
Y +
Yn
n =
Y + n−1 otherwise,
if Yn ≥ qα,