What Is the Limit of a Sum of Cauchy Random Variables?

The phrase “limit of a sum” can refer to the unnormalized sum, the sample mean, or convergence in distribution. For independent Cauchy random variables, these have different answers from the familiar results for finite-variance data.

Assume first that $X_1,X_2,\ldots$ are independent and identically distributed standard Cauchy variables. Their sum is Cauchy with a scale that grows with the number of terms. But their sample mean has exactly the same standard Cauchy distribution at every sample size: it does not settle near a fixed number.

The Cauchy distribution and its scale

A Cauchy variable with location $\mu$ and scale $\gamma>0$, written $\operatorname{Cauchy}(\mu,\gamma)$, has density

$$f(x)=\frac{1}{\pi\gamma\left[1+\left(\frac{x-\mu}{\gamma}\right)^2\right]}.$$

The standard Cauchy distribution has $\mu=0$ and $\gamma=1$. Its characteristic function is

$$\varphi_X(t)=\mathbb{E}[e^{itX}]=e^{i\mu t-\gamma|t|}.$$

The scale $\gamma$ plays a role similar to a measure of spread, but it is not a standard deviation: a Cauchy variable has no finite mean or variance. In particular, the ordinary law of large numbers, which requires a finite expected absolute value in its usual form, cannot be applied to Cauchy observations.

Sum of independent Cauchy variables

Let $X_1,\ldots,X_n$ be independent, with $X_j\sim\operatorname{Cauchy}(\mu_j,\gamma_j)$. Independence means the characteristic function of their sum is the product of the individual characteristic functions:

$$\begin{aligned} \varphi_{X_1+\cdots+X_n}(t) &=\prod_{j=1}^n e^{i\mu_jt-\gamma_j|t|}\\ &=e^{i(\sum_j\mu_j)t-(\sum_j\gamma_j)|t|}. \end{aligned}$$

This is the characteristic function of a Cauchy distribution. Therefore,

$$X_1+\cdots+X_n\sim\operatorname{Cauchy}\left(\sum_{j=1}^n\mu_j,\sum_{j=1}^n\gamma_j\right).$$

For iid $X_j\sim\operatorname{Cauchy}(\mu,\gamma)$, this becomes

$$S_n=X_1+\cdots+X_n\sim\operatorname{Cauchy}(n\mu,n\gamma).$$

So the unnormalized sum does not converge to a finite random variable as its scale grows. In the centered standard case, $S_n\sim\operatorname{Cauchy}(0,n)$; for any fixed $M>0$,

$$\mathbb{P}(|S_n|\le M)=\frac{2}{\pi}\arctan\left(\frac{M}{n}\right)\longrightarrow 0.$$

Thus $|S_n|$ tends to infinity in probability in this centered case. This does not mean the sum grows steadily in one direction: its values can be positive or negative, with increasingly broad spread.

What happens to the sample mean?

For iid Cauchy variables, define $\overline X_n=S_n/n$. Scaling a Cauchy variable by a positive constant scales its location and scale by that constant. Hence

$$\overline X_n\sim\operatorname{Cauchy}(\mu,\gamma)$$

for every $n$. In the standard case, the distribution of the average is exactly the standard Cauchy distribution, regardless of how many observations you average.

For example, if $X_1,\ldots,X_{100}$ are iid standard Cauchy, then $\overline X_{100}$ is still standard Cauchy. Its probability of lying outside $[-10,10]$ is

$$\mathbb{P}(|\overline X_{100}|>10) =1-\frac{2}{\pi}\arctan(10) \approx 0.0635.$$

Collecting more observations has not made the average concentrate around zero. The mean is zero as a symmetry center, but the Cauchy expectation is undefined, so it is not a finite expected value to which the sample mean could converge.

Convergence in distribution is different

Because every $\overline X_n$ has the same Cauchy distribution, the sequence $\overline X_n$ converges in distribution to a Cauchy variable with that distribution. This statement concerns the distributions of the averages; it does not say that the averages from one growing data set settle down to a value.

In fact, the averages do not converge in probability for any iid Cauchy$(\mu,\gamma)$ variables with $\gamma>0$. To see why, compare $\overline X_{2n}$ with $\overline X_n$:

$$\overline X_{2n}-\overline X_n =\frac{(X_{n+1}+\cdots+X_{2n})-(X_1+\cdots+X_n)}{2n}.$$

The two block sums are independent Cauchy variables, each with location $n\mu$ and scale $n\gamma$. Their difference has location $0$ and scale $2n\gamma$, and division by $2n$ gives a $\operatorname{Cauchy}(0,\gamma)$ variable. Consequently, for every $\varepsilon>0$,

$$\mathbb{P}(|\overline X_{2n}-\overline X_n|>\varepsilon) =1-\frac{2}{\pi}\arctan\left(\frac{\varepsilon}{\gamma}\right),$$

which is positive and does not depend on $n$. A sequence that converges in probability must have differences between such late terms tending to zero in probability, so this rules out convergence in probability. Almost-sure convergence is ruled out as well, since it implies convergence in probability.

Common mistakes

• Assuming the central limit theorem applies. The usual central limit theorem requires finite variance. Cauchy variables have no finite variance, and their normalized sums do not approach a normal distribution.
• Treating the Cauchy location as an expectation. The location parameter describes the distribution’s center and median, but the expectation is undefined.
• Confusing distributional stability with sample-path convergence. The sample mean’s distribution stays fixed, but that does not mean a particular sequence of observed averages converges.
• Ignoring dependence. The sum rule above uses independence. If the variables are dependent, the distribution of their sum need not follow from adding the individual scales.

The practical rule is to state both the assumptions and the type of limit: independent Cauchy sums have additive scales; iid Cauchy sample means keep their original distribution and do not satisfy the usual law of large numbers.