Research proof index

Any uncountable set in a separable, densely-ordered, first-countable linear order admits a strictly decreasing sequence of its elements converging to a point from the right.

A stochastic process X with independent increments and such that X t is gaussian for all t is a Gaussian process.

Two internally independent families remain jointly independent after they are combined over a disjoint union, provided the two family-valued random variables are independent of one another.

Two-sided expectation bound for adapted {0,1}-weighted increment sums of X, from the boundedness of elementary stochastic integrals at time t. The lower bound uses the complementary weights 1 - W.

Blumenthal's zero-one law: Let 𝓕 be the canonical filtration associated to a Brownian motion. Then the σ-algebra ⨅ s > 0, 𝓕 s is trivial.

A locally square-integrable martingale has locally submartingale squared norm.