Sufficiency in sequentially planned decision procedures

Abstract

Sampling plan, sequentially planned procedure, sufficient σ-fields, 62L10, 60G40,

The optional sampling theorem for submartingales in the sequentially planned context

“…Proof: {B τn } n∈N is a filtration since τ n τ n+1 (see Fenoy and Ibarrola (2003)). For B ∈ B τn , it holds that…”

“…Z a , is a uniformly integrable martingale. Let τ be a sampling plan; we consider the admissible sampling plans that we have defined in Definition 3.1, that is, k τ , ∀k ∈ N. By the optional projection theorem (see Fenoy and Ibarrola (2003)),…”

“…the submartingales are indexed by a set A of sampling sequences, and a partial order represented by is defined on A. Thus, let a = (a 1 , · · · , a j ) ∈ A, b = (b 1 , · · · , b m ) ∈ A finite sampling sequences; a b means that a 1 = b 1 , · · · , a j = b j with j ≤ m, and each a i , b k ∈ M ⊂ N. For details, see Schmitz (1993), Fenoy and Ibarrola (2003).…”

“…We do not discuss some concepts in detail since the background of this note can be found in the reference Fenoy and Ibarrola (2003).…”