# Law of large numbers: other formulae

Let $$X_1, X_2, \ldots$$ be an infinite sequence of i.i.d. random variables with $$E(X_i)=\mu$$ and $$\mbox{Var}(X_i) < \infty$$.

The law of large numbers states $$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{X_i}{n} = \mu$$.

Wikipedia mentions that "other formulas that look similar are not verified".

In other words $$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} X_i = n\mu$$ is not correct?

Is there a counter-example or an obvious reason why this is not the case?

• Note also that the Law of Large Numbers does not require the existence of the variance of the $X_i$'s. Commented Oct 13, 2022 at 6:49
• A warning that arises every time in the classroom:$$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} X_i = n\mu$$does not make sense (mathematically) since the lhs does not depend on $n$ as a limit, while the rhs does. Commented Oct 13, 2022 at 6:50
• No it does not make sense either: $\mathbb E(X_i)=\mu/n$ has the lhs depending on $i$ and the rhs depending on $n$. Commented Oct 13, 2022 at 7:23
• Sorry, but the fundamental reason is that $\mathbb E(X_i)=\mu/n$ does not make sense mathematically. The random variable $X_i$ for one given $i\in\mathbb N^*$, does not depend on $n$. Commented Oct 13, 2022 at 10:12
• In that case, you need to rephrase the question differently. It sounds like you are dealing with the law of large numbers for triangular arrays. Commented Oct 14, 2022 at 14:10

What do LLNs say?

At the risk of getting too oversimplistic, for independent sequence $$\langle X_i\rangle_{i\in\mathbb N},$$ they say the partial sum (or average, precisely)

$$\frac{1}{n}(X_1+X_2+\ldots+ X_n) \tag 1$$

converges to common mean $$\mu$$ in certain sense provided second moment or forth moment be finite and uniformly bounded depending on what convergence one is bothered with.

When $$\rm i.i.d.$$ is imposed, only the first moment needs to be finite.

What is Wikipedia saying?

It is only saying form like

$$\sum_{i=1}^n X_i -n\times \bar X$$

doesn't make sense for

not only does it not converge toward zero as $$n$$ increases, but it tends to increase in absolute value as $$n$$ increases.

Finally as Xi'an noted in the comments, be careful with indexing set and limiting variable.

$$\rm [I]$$ A First Look at Rigorous Probability Theory, Jeffrey S. Rosenthal, World Scientific Publishing, $$2006,$$ section $$5.3,~5.4,$$ pp. $$60-64.$$