Confidence sequence for a time-varying target

Let $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbb{N}}_0},P)$ be a discrete filtered probability space and let $({\theta }_t{)}_{t\ge 1}$ be a real target process with each ${\theta }_t$ measurable with respect to ${\mathcal{F}}_t$. For $\alpha \in (0,1)$, a sequence of random closed intervals $C_t=[L_t,U_t]$, with finite real ${\mathcal{F}}_t$-measurable endpoints at each $t\ge 1$, is a $(1-\alpha )$ confidence sequence for the target process $({\theta }_t)$ if $P\text{\hspace{0.17em}⁣}({\theta }_t\in C_t\text{ for every }t\ge 1)\ge 1-\alpha \mathrm{.}$ The target may be random and may vary with time; the definition asserts only simultaneous coverage under the specified probability law.

Filtration and filtered probability space: the filtration is an increasing sequence of sigma-algebras on one probability space, indexed from time zero.

Adapted and integrable stochastic process: measurability of each ${\theta }_t$ with respect to ${\mathcal{F}}_t$ is the adaptedness condition at positive times.

A Howard et al., §2, including the changing average conditional-mean target