An Introduction to Probability Theory and Its Applications, Volume 2

a₁ applies arbitrary argument assume asymptotic atoms backward equation Baire functions Borel sets bounded central limit theorem characteristic function common distribution compound Poisson condition consider continuous function convergence convolution defined definition denote derived distribution concentrated distribution F distribution function equals example exists exponential distribution F{dx finite interval fixed follows formula Fourier given hence implies independent random variables inequality infinitely divisible integral Laplace transform law of large left side lemma Let F limit distribution limit theorem Markov martingale measure mutually independent normal distribution notation o-algebra operator parameter Poisson process positive probabilistic probability distribution problem proof prove random walk renewal epochs renewal equation renewal process S₁ sample space satisfies semi-group sequence shows solution stable distributions stochastic stochastic kernel symmetric T₁ tends theory transition probabilities uniformly unique variance vector X₁ Y₁ zero expectation