Conditional weak laws in Banach spaces

Abstract:

Let $(B,\|\cdot \|)$ be a separable Banach space. Let $Y, Y_1, Y_2, \ldots$ be centered i.i.d. random vectors taking values on $B$ with law $\mu$, $\mu (\cdot )=P(Y\in \cdot )$, and let $S_n =\sum _{i=1}^n Y_i.$ Under suitable conditions it is shown for every open and convex set $0 \notin D\subset B$ that $P\left ( \|{\frac {{\displaystyle S_n}}{\displaystyle n}} - v_d \|> \varepsilon \Big |\frac {{\displaystyle S_n}}{\displaystyle n}\in D\right )$ converges to zero (exponentially), where $v_d$ is the dominating point of $D.$ As applications we give a different conditional weak law of large numbers, and prove a limiting aposteriori structure to a specific Gibbs twisted measure (in the direction determined solely by the same dominating point).