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If ${\sup }_n\mathbb{E}|X_n{|}^2<\infty$ and the $X_n$ are pairwise uncorrelated random variables, then $n^{-1}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)\to 0$ almost surely. Independence is unnecessary for this version of the strong law of large numbers. A uniformly bounded martingale difference sequence meets its hypotheses, since the differences have mean zero and are orthogonal in $L^2$.

Two square-integrable random variables $X,Y$ are uncorrelated random variables when $\mathrm{Cov}(X,Y)=0$. Independent random variables with finite second moments are uncorrelated random variables, but zero covariance does not generally imply independence.

For an explicit counterexample, let $X$ take $-1$, $0$, $1$ with equal probabilities, and let $Y=X^2$. Then $\mathbb{E}X=\mathbb{E}{X}^3=0$, so $\mathrm{Cov}(X,Y)=\mathbb{E}{X}^3-\mathbb{E}X\mathbb{E}{X}^2=0$. Nevertheless, $Y$ is completely determined by $X$: $\mathbb{P}(Y=0\mid X=0)=1$, while $\mathbb{P}(Y=0)=1/3$. Thus these are uncorrelated random variables with statistical dependence. In this example, opposite contributions from the odd function $x^3$ cancel; the even function $x^2$ still retains information about $|X|$. The last panel of the covariance diagram shows the relationship.

There are useful special cases. Uncorrelated jointly normal variables are independent, and the Bernoulli independence from zero covariance result supplies another setting where the converse holds. Their distribution assumptions do the extra work; zero covariance alone does not.

A white noise sequence has zero mean, a common finite variance and zero autocovariance at every nonzero lag. This requires uncorrelated random variables, rather than independent random variables.

For the AR(1) and MA(1) formulas here, white noise means a real sequence indexed by all integer times withThese are the weak white noise assumptions: zero mean, a common finite positive variance, and uncorrelated random variables at distinct times. Independence and a normal distribution are additional assumptions. Strong white noise requires independent and identically distributed random variables; discrete Gaussian white noise is the jointly Gaussian special case. Some definitions allow a constant nonzero mean; subtract it before using the centered formulas here.

For a concrete dependent example, let $A$ equal $1$ or $2$ with equal probabilities, independently of a sequence of independent Rademacher random variables $Z_t$, and put ${\epsilon }_t=A{Z}_t$. Its mean is zero, its variance is $\mathbb{E}[A^2]=5/2$, and its nonzero-lag autocovariances vanish. Nevertheless every $|{\epsilon }_t|$ equals the same $A$: seeing one magnitude determines all the others. This is white noise with dependence invisible to its ACF, and is also an example of the issue discussed in ergodic stationary process.