In discussions of maximization of expected value, the Law of Large
Numbers is sometimes invoked, at times—especially by me—off-handedly.
According to the Strong Law of Large Numbers (SLLN), if you have an
infinite sequence of independent random variables *X*_{1}, *X*_{2}, ...
satisfying some conditions (e.g., in the Kolmogorov version ∑_{n}(*σ*_{n}^{2}/*n*^{2}) < ∞,
where *σ*_{n}^{2}
is the variance of *X*_{n}), then with
probability one, the average of the random variables converges to the
average of the mathematical expectations of the random variables. The
thought is that in that case, if the expectation of each *X*_{n} is positive,
it is rationally required to accept the bet represented by *X*_{n}.

In a
recent post, showed how in some cases where the Strong Law of Large
Numbers is not met, in an infinite run it can be disastrous to bet in
each case according to expected value.

Here I want to make a minor observation. The fact that the SLLN
applies to some sequence of independent random variables is itself not
sufficient to make it rational to bet in each case according to the
expectations in an infinite run. Let *X*_{n} be 2^{n}/*n* with
probability 1/2^{n}
and − 1/(2*n*) with probability
1 − 1/2^{n}. Then

*E**X*_{n} = (1/2^{n})(2^{n}/*n*) − 1/(2*n*)(1−1/2^{n}) = (1/*n*)(1−(1/2)(1−1/2^{n})).

Clearly *E**X*_{n} > 0.
So in individual decisions based on expected value, each *X*_{n} will be a
required bet.

Now, just as in my previous post, almost surely (i.e., with
probability one) only finitely many of the bets *X*_{n} will have the
positive payoff. Thus, with a finite number of exceptions, our sequence
of payoffs will be the sequence − 1/2, − 1/4, − 1/6, − 1/8, ....
Therefore, almost surely, the average of the first *n* payoffs converges to zero.
Moreover, the average of the first *n* mathematical expectations
converges to zero. Hence the variables *X*_{1}, *X*_{2}, ...
satisfy the Strong Law of Large Numbers. But what is the infinite run
payoff of accepting all the bets? Well, given that almost surely there
are only a finite number of *n*
such that the payoff of bet *n*
is not of the form − 1/(2*n*),
it follows that almost surely the infinite run payoff differs by a
finite amount from − 1/2 − 1/4 − 1/6 − 1/8 = − ∞. Thus the
infinite run payoff is negative infinity, a disaster.

Hence even when the SLLN applies, we can have cases where almost
surely there are only finitely many positive payments, infinitely many
negative ones, and the negative ones add up to − ∞.

In the above example, while the variables satisfy the SLLN, they do
not satisfy the conditions for the Kolmogorov version of the SLLN: the
variances grows exponentially. It is somewhat interesting to ask if the
variance condition in the Kolmogorov Law is enough to prevent this
pathology. It’s not. Generalize my example by supposing that *a*_{1}, *a*_{2}, ...
is a sequence of numbers strictly between 0 and 1 with
finite sum. Let *X*_{n} be 1/(*n**a*_{n})
with probability *a*_{n} and − 1/(2*n*) with probability 1 − *a*_{n}. As
before, the expected value is positive, and by Borel-Cantelli (given
that the sum of the *a*_{n} is finite)
almost surely the payoffs are − 1/(2*n*) with finitely many
exceptions, and hence the there is a finite positive payoff and an
infinite negative one in the infinite run.

But the variance *σ*_{n}^{2}
is less than *a*_{n}/(*n**a*_{n})^{2} + 1 = (1/(*n*^{2}*a*_{n})) + 1.
If we let *a*_{n} = 1/*n*^{2}
(the sum of these is finite), then each variance is at most 2, and so the conditions of the Kolmogorov
version of the SLLN are satisfied.

In an
earlier post, I suggested that perhaps the Central Limit Theorem
(CLT) rather than the Law of Large Numbers is what one should use to
justify betting according to expected utilities. If the variables *X*_{1}, *X*_{2}, ...
satisfy the conditions of the CLT, and have non-negative expectations,
then *P*(*X*_{1}+...+*X*_{n}≥0)
will eventually exceed any number less than 1/2. In particular, we won’t have the kind of
disastrous situation where the overall payoffs almost surely go
negative, and so no example like my above one can satisfy the conditions
of the CLT.