# The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia.

The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak law. The question is then: what is the reason for keeping them separated? Why do they always presented as two distinct results? If we assume the same, why would we wish to restrain what we get? Probably, I just do not really realize how the two laws are being used.

Also, there is the following statement under Weak law:

Convergence in probability is also called weak convergence of random variables.

Is not it about convergence in distribution?

Thank you.

Regards, Ivan

• For one, the weak law is much easier to prove than the strong law (it's just a consequence of Chebyshev's inequality at least when finite variance is assumed). Jun 3, 2013 at 9:06
• In addition to the ease of proof, it may have to do with the history. The Weak Law was proven by Bernoulli in the 18th century. The Strong Law was proven by Borel in the beginning of the 20th century. Jun 3, 2013 at 10:17
• See here: math.stackexchange.com/questions/13421/…. I am not 100% sure but it seems that if a sequence of random variables satisfy the hypotheses of either the WLLN or the SLLN then both results hold (since the hypotheses are the same). Conversely if you do not assume the hyptotheses of the WLLN (or the SLLN) hold then theoretically you might be able to contrive an example whereby a sequence of random variables converges in probability but not almost surely to a particular value. Such an example is contained in the link. Jun 5, 2013 at 8:18

Weak laws are about convergence in probability and not convergence in distribution. Convergence in probability implies convergence in distribution but not the other way round.

One advantage of the weak law is that there need not be one probability space over which the entire sequence of random variables is simultaneously defined.

Thus suppose you want to formalise the statement “when tossing an unbiased coin about half of the tosses will be heads”. The weak law version can be proved in the context of elementary discrete probability by considering a sequence of probability space where a sample point in the $n$-th space consists of the outcome of the first $n$ tosses. Then by Chebyshev's inequality applied in this $n$-th space $$P(|S_n/n-1/2|>\epsilon)\le 1/(4n\epsilon^2)$$ and an elementary limit gives the weak law.

In the same setting the strong law would require us to consider infinite sequences of tosses as outcomes and that is not possible without using the tools of measure theory.

And sometimes it is not only technical convenience but the nature of the question itself which requires us to use different probability spaces for different values of $n$. see Example 6.3 in Billingsley's Probability and Measure.

• Regarding my last question and your first statement, the terminology used in Wikipedia is not entirely clear for me. I quoted a sentence from Law of large numbers which says that convergence in probability is called weak convergence. At the same time, according to Convergence of random variables, "converge in distribution" is also referred to as "converge weakly." So, what does the word weak belong to? Thank you!
– Ivan
Dec 7, 2013 at 9:58