As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first "n" values goes to zero as "n" goes to infinity. As an example, assume that each random variable in the series follows a Gaussian distribution with mean zero, but with variance equal to formula_8, which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to formula_9. The variance of the average is therefore asymptotic to formula_10 and goes to zero.