The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

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Principles Of Random Walk

: In the study of real-valued representations (Paretian utility functions) of a completely preordered space (Space of actions preordered by preferences), the continuity properties of the representation can be made to rest on the following result. R will denote the extended real line, i.e., the real line with negative infinity and positive infinity adjoined. A degenerate set is a set having at most one element. Given a subset S of R, a lacuna of S is a non-degenerate interval of R without points of S but having a lower bound and an upper bound in S; a gap of S is a maximal lacuna of S. (Author)

On the continuity properties of paretian utility

We modify the principal-agent model with moral hazard by assuming that the agent is expectation-based loss averse according to Koszegi and Rabin (2006, 2007). The optimal contract is a binary payment scheme even for a rich performance measure, where standard preferences predict a fully contingent contract. The logic is that, due to the stochastic reference point, increasing the number of different wages reduces the agent’s expected utility without providing strong additional incentives. Moreover, for diminutive occurrence probabilities for all signals the agent is rewarded with the fixed bonus if his performance exceeds a certain threshold.

Binary Payment Schemes: Moral Hazard and Loss Aversion

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the \peano curve" surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a p 4=3-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter = 12 (i.e., SLE12). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, k-angulations, and maps in which face sizes are mixed.

Bipolar orientations on planar maps and SLE 12

Random walks in cones: The case of nonzero drift

Invariance principles for random walks in cones

We prove invariance principles for a mulditimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of $h$-transformed random walk to the corresponding $h$-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

Martin boundary of random walks in convex cones

We characterize in terms of primitives on risk preferences the stopping and the continuation regions of optimal stopping problems, where an agent without commitment decides when to stop a diffusion. We consider separately the case of a naive agent who is unaware of the possible time inconsistency and the case of a sophisticated agent who is fully aware of a possible time inconsistency in her behavior. Among other things, we apply the general results to fully characterize the extreme cases of never stopping for the naive and never starting for the sophisticate across an extensive range of non-expected risk preference models.

Jetlir Duraj

Papers

Random walks in cones: The case of nonzero drift

Invariance principles for random walks in cones

Invariance principles for random walks in cones

Martin boundary of random walks in convex cones

Optimal Stopping with General Risk Preferences