Research proof index

{τₙ(c) < 1} = {c < Aⁿ₁}.

A family of random variables is uniformly integrable iff its L¹ tails above c tend to zero uniformly in the index.

An analogue of uniformIntegrable_iff, but the nonstrict inequality in C ≤ ‖X i x‖₊ is replaced by a strict inequality.

Multidistance is unchanged when a finite family of random variables is reindexed along an equality $m'=m$. This says that the quantity depends on the family, not on the particular equal presentation of its finite index type.

Let m ≥ 2, and let X_[m] be a tuple of G-valued random variables. Let W := ∑ X_i. Then d[W;-W] ≤ 2 D[X_i].

If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$.