For any sequence {Xi}, the sequence Sk(X) = Sk := X1 + · · · + Xk is known as the sequence of partial sums. We study the behaviour of this and related sequences in this chapter, when {Xi} are independent. We first discuss the strong law of large numbers which roughly says that the time average of iid random variables converges a.s. to the space average, as the number of summands goes to 1. This extends the weak law of large numbers presented in Exercise 14.2.8. Then we study the probabilistic behaviour of max1_k_n Sk as compared to that of Sn. These are known as maximal inequalities. Then we move to the study of the behaviour of partial sums as n ! 1. In particular, for independent rvs, Sn converges a.s. if and only if it converges in probability. The celebrated Kolmogorov three series theorem which relates the convergence of partial sums to the convergence of the sum of means, variance and tail probabilities of related rvs is established. We state without proof the law of iterated logarithm which gives the behaviour of the limit points of appropriately scaled partial sums. Finally, we present a few important inequalities, which substantially improve the polynomial upper bound in Markov’s inequality Lemma 8.1.1, to exponential bounds. In particular, we cover Hoeffding’s inequality and, without proof, an inequality of Talagrand.