Rewriting the variance Sample Clauses

Rewriting the variance. In order to obtain the desired lower bound on βpjj (t), we need to obtain a lower µp,t bound for the variance σ2 (−βpj (t) log |T j| + fp). From now on we shall denote fp,t : [0, 1] → R ∪ {∞} by fp,t = −βpj (t) log |T j| + fp. (4.7) p,t ∫ This section is dedicated to rewriting σ2 (fp,t) in a way that will allow us to estimate n it from below. Note that fp,tdµp,t = 0 by (4.5). Recall that by (2.13) and the fact that the variance is invariant under adding a coboundary, it follows that σ p,t (fp,t ) = ∫ ˜2 p,t dµp,t ∞ Σ .∫ + 2 n=1 f˜p,t · f˜p,t ◦ Tp dµ p,t Σ (4.8) for any function f˜p,t which is cohomologous to fp,t. The second term on the right hand side of (4.8) is what makes it difficult to study lower bounds on the variance. Therefore, our aim is to find a function f˜p,t which is cohomologous to fp,t, that is, a coboundary Up,t − Up,t ◦ Tp such that f˜p,t = fp,t + Up,t − Up,t ◦ Tp, for which the right hand term in (4.8) vanishes. Therefore, in the first part of this section we introduce a family of transfer operators which will aid us towards obtaining the p,t