A Weak Law of Large Numbers is a proposition stating a set of conditions guaranteeing that the mean of a sample converges in probability - as the sample size increases - to the true mean of the probability distribution from which the sample has been extracted.
The adjective weak is used because convergence in probability is often called weak convergence, and it is employed to make a distinction from Strong Laws of Large Numbers, in which the sample mean is required to converge almost surely.
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The acronym WLLN is frequently used as an abbreviation of Weak Law of Large Numbers.
A more precise definition follows.
Definition
Let
![[eq1]](/images/Weak_Law_of_Large_Numbers__1.png){kind=small}
be a sequence of random variables with mean
![[eq2]](/images/Weak_Law_of_Large_Numbers__2.png){kind=small}
.
Then, we say that a Weak Law of Large Numbers holds if and only
if![[eq3]](/images/Weak_Law_of_Large_Numbers__3.png)
where
denotes a limit in probability.
The conditions that need to be imposed on the sequence
![[eq4]](/images/Weak_Law_of_Large_Numbers__5.png){kind=small}
in order to derive a Weak Law are usually quite mild. For example, in
Chebyshev's WLLN, it is only required that
be covariance stationary and that
auto-covariances between the terms of the sequence be on average equal to
zero.
The lecture entitled Laws of Large Numbers discusses Weak and Strong Laws in more detail. In particular, it provides precise statements and proofs of two versions of Chebyshev's WLLN.
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