On the contraction method with degenerate limit equation

A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the ``unique'' solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000017

On a functional contraction method

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space $\mathcal{C}[0,1]$ of continuous functions endowed with uniform topology and the space $\mathcal {D}[0,1]$ of c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach's fixed-point theorem. We develop the use of the Zolotarev metrics on the spaces $\mathcal{C}[0,1]$ and $\mathcal{D}[0,1]$ in this context. Applications are given, in particular, a short proof of Donsker's functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.Comment: Published at http://dx.doi.org/10.1214/14-AOP919 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sequential Estimation of Structural Models with a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze the conditions under which the nested pseudo-likelihood (NPL) algorithm achieves convergence and derive its convergence rate. We find that the NPL algorithm may not necessarily converge when the fixed point mapping does not have a local contraction property. To address the issue of non-convergence, we propose alternative sequential estimation procedures that can achieve convergence even when the NPL algorithm does not. Upon convergence, some of our proposed estimation algorithms produce more efficient estimators than the NPL estimator.contraction, dynamic games, nested pseudo likelihood, recursive projection method

Sequential Estimation of Structural Models with a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze the conditions under which the nested pseudo-likelihood (NPL) algorithm achieves convergence and derive its convergence rate. We find that the NPL algorithm may not necessarily converge when the fixed point mapping does not have a local contraction property. To address the issue of non-convergence, we propose alternative sequential estimation procedures that can achieve convergence even when the NPL algorithm does not. Upon convergence, some of our proposed estimation algorithms produce more efficient estimators than the NPL estimator.contraction, dynamic games, nested pseudo likelihood, recursive projection method

Notes on the applicability of contraction method for stable limit laws

We presented a proof for the classical stable limit laws under use of contraction method in combination with the Zolotarev metric. Furthermore, a stable limit law was proved for scaled sums of growing into sequences. This limit law was alternatively formulated for sequences of random variables defined by a simple degenerate recursion