i Robin Pemantle’s Lecture Notes Cornell Summer School in Probability 6 July – 17 July, 2009 Contents 1 Fourier-Laplace integrals 1-1 1.1Laplace-typeintegralsinonevariable...................... 1-2 1.2Laplace-typeintegralsinseveralvariables................... 1-4 1.3Complexphases.................................. 1-7 2 Generating functions 2-1 2.1Formalpowerseriesandcombinatorialenumeration............. 2-1 2.2Applicationsinprobability........................... 2-5 2.3Coefficientextractionandtransfertheorems.................. 2-8 3LLDandLCLT 3-1 3.1Thesmoothpointtheorem........................... 3-2 3.2Locallargedeviations.............................. 3-4 3.3LocalCLT..................................... 3-7 3.4Smoothpointapplications............................ 3-8 5 Proofs 5-1 5.1Firstproof..................................... 5-1 5.2Secondproofandcanonicalrepresentations.................. 5-3 5.3Non-smoothpoints-asketchoffurtherresults................ 5-6 iii Bibliography [ABG70] M. 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