## MODES OF CONVERGENCE

Let us first review the definition of the convergence of a sequence of real numbers.

DEFINITION 6.1.1 Asequence of real numbers {aj, n = 1, 2, . . . , is said to converge to a real number a if for any e > 0 there exists an integer N such that for all n > N we have

(6.1.1) |an — a| < e.

We write a„ —» a as n —> °° or lim,,^» an = a (n —> °° will be omitted if it is obvious from the context).

Now we want to generalize Definition 6.1.1 to a sequence of random variables. If a„ were a random variable, we could not have (6.1.1) exacdy, because it would be sometimes true and sometimes false. We could only talk about the probability of (6.1.1) being true. This suggests that we should modify the definition in such a way that the conclusion states that

(6.1...

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