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null (Ed.)We study a sequence of many-agent exit time stochastic control problems, parameterized by the number of agents, with risk-sensitive cost structure. We identify a fully characterizing assumption, under which each such control problem corresponds to a risk-neutral stochastic control problem with additive cost, and sequentially to a risk-neutral stochastic control problem on the simplex that retains only the distribution of states of agents, while discarding further specific information about the state of each agent. Under some additional assumptions, we also prove that the sequence of value functions of these stochastic control problems converges to the value function of a deterministic control problem, which can be used for the design of nearly optimal controls for the original problem, when the number of agents is sufficiently large.more » « less
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null (Ed.)Abstract The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓ p -spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓ p -spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓ p -balls in a high-dimensional Euclidean space.more » « less
