Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Berry Esseen type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds are applied to the occurrences of fixed relatively ordered sub-sequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs on finite graphs.Comment: 23 page

Epistemic Modal Credence

Triviality results threaten plausible principles governing our credence in epistemic modal claims. This paper develops a new account of modal credence which avoids triviality. On the resulting theory, probabilities are assigned not to sets of worlds, but rather to sets of information state-world pairs. The theory avoids triviality by giving up the principle that rational credence is closed under conditionalization. A rational agent can become irrational by conditionalizing on new evidence. In place of conditionalization, the paper develops a new account of updating: conditionalization with normalization

Normal approximation for hierarchical structures

Given F:[a,b]^k\to [a,b] and a nonconstant X_0 with P(X_0\in [a,b])=1, define the hierarchical sequence of random variables {X_n}_{n\ge 0} by X_{n+1}=F(X_{n,1},...,X_{n,k}), where X_{n,i} are i.i.d. as X_n. Such sequences arise from hierarchical structures which have been extensively studied in the physics literature to model, for example, the conductivity of a random medium. Under an averaging and smoothness condition on nontrivial F, an upper bound of the form C\gamma^n for 0<\gamma<1 is obtained on the Wasserstein distance between the standardized distribution of X_n and the normal. The results apply, for instance, to random resistor networks and, introducing the notion of strict averaging, to hierarchical sequences generated by certain compositions. As an illustration, upper bounds on the rate of convergence to the normal are derived for the hierarchical sequence generated by the weighted diamond lattice which is shown to exhibit a full range of convergence rate behavior.Comment: Published at http://dx.doi.org/10.1214/105051604000000440 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org