This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Khler-Einstein metric. The main result of this paper is that a Khler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Khler metrics with controlled geometry in the Gromov-Hausdorff sense.

This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Khler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi . We also put all our technical results together to complete the proof of the main theorem.

When people in a society want to make inference about some parameter, each person would potentially want to use data collected by other people. Information (data) exchange in social contexts is usually costly, so to make sound statistical decisions, people need to compromise between benefits and costs of information acquisition. Conflicts of interests and coordination will arise. Classical statistics does not consider peoples interaction in the data collection process. To address this ignorance, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Bearing our interest in aggregate inference at the societal level, we propose a new concept finite population learning to address whether with high probability, a large fraction of people can make good inferences. Serving as a foundation, this concept enables us to study the long run trend of aggregate inference quality as population grows.

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Economists historically measure the degree to which the market is surprised by an earnings announcement by the consensus forecast error, defined as difference between the actual earnings and the consensus forecast. The consensus might be calculated using either the mean or median of security analysts forecasts. The premise of this measure is that the consensus forecast is a good proxy for the markets expectation of earnings. Hence the consensus forecast error captures how surprised the market is when the earnings is announced. The consensus forecast error is a building block of a host of studies across finance, accounting and economics (see for a survey of event studies in Kothari (2001)). For instance, in finance and accounting, it is used in event studies of how efficiently markets react to earnings announcements. Efficient market studies when it comes to bond or currency markets and macroeconomic