Why a Cauchy Criterion for Random Variables?
To show that a sequence of real numbers converges, one can often verify that its terms eventually become arbitrarily close to one another, without first identifying the limit. The same idea works for random variables, but the statement must be made sample path by sample path: for almost every outcome, the realized sequence of numbers must be Cauchy.
This criterion is useful when a sequence has a natural pathwise structure but its limit is not yet known. There is also a measurability issue to settle: the event that a path is Cauchy involves infinitely many indices and tolerances. Expressing the tolerances with a countable set makes that event measurable and gives an exact formulation of the criterion.
The Sample-Path Cauchy Event
Let \((\Omega,\mathcal F,P)\) be a probability space, and let \(X_1,X_2,\ldots\) be real-valued random variables on it. For a fixed \(\omega\in\Omega\), the sequence \(X_1(\omega),X_2(\omega),\ldots\) is a sequence of real numbers. It is Cauchy if, for every \(\varepsilon>0\), there is an integer \(N\) such that all terms with indices at least \(N\) differ by less than \(\varepsilon\).
The displayed expression uses only countably many tolerances, namely \(1,1/2,1/3,\ldots\). This is enough for the full Cauchy condition. Given any \(\varepsilon>0\), choose an integer \(m\) large enough that \(1/m<\varepsilon\). If the terms are eventually within \(1/m\) of one another, they are also eventually within \(\varepsilon\).
The event \(C\) is measurable. For each fixed \(j,k\), the difference \(X_j-X_k\) is measurable, so the event \(\{|X_j-X_k|<1/m\}\) belongs to \(\mathcal F\). The intersections over \(j,k\geq N\), the unions over \(N\), and the intersection over \(m\) are all countable operations. Thus \(C\in\mathcal F\).
Worked Example: A Random Sequence with a Known Pathwise Limit
Let \(U\) be uniformly distributed on \([0,1]\), and set \(X_n=U+1/n\). Verify the Cauchy property and identify the limit.
For any outcome \(\omega\) and any \(j,k\geq N\),
For any integer \(m\geq1\), choose \(N\) so that \(1/N<1/m\). Then every pair \(j,k\geq N\) satisfies \(|X_j(\omega)-X_k(\omega)|<1/m\), for every \(\omega\). Hence \(C=\Omega\) and \(P(C)=1\). Directly, \(X_n(\omega)\to U(\omega)\) for every outcome.
The Almost-Sure Cauchy Criterion
Proof. First suppose that \(X_n\to X\) almost surely for a real-valued random variable \(X\). There is an event \(G\in\mathcal F\) with \(P(G)=1\) such that \(X_n(\omega)\to X(\omega)\) for every \(\omega\in G\). For each such \(\omega\), a convergent sequence of real numbers is Cauchy: given \(\varepsilon>0\), choose \(N\) such that \(|X_j(\omega)-X(\omega)|<\varepsilon/2\) for every \(j\geq N\). Then for \(j,k\geq N\), the triangle inequality gives
Thus \(G\subseteq C\), so \(P(C)=1\).
Conversely, suppose \(P(C)=1\). For every \(\omega\in C\), the real sequence \(X_n(\omega)\) is Cauchy. Completeness of \(\mathbb R\) implies that it converges to a finite real number. To make this pathwise limit a random variable on all of \(\Omega\), define
Each \(Z_n\) is measurable because \(C\) is measurable and \(X_n\) is measurable. On \(C\), the sequence \(Z_n(\omega)\) converges by completeness of \(\mathbb R\); on \(C^c\), every \(Z_n(\omega)\) equals zero. Thus \(Z_n\) converges everywhere. Its pointwise limit \(X\) is measurable. On \(C\), \(Z_n=X_n\), so \(X_n\to X\) there. Since \(P(C)=1\), this is almost-sure convergence to a finite real-valued random variable. \(\square\)
The construction of \(X\) outside \(C\) is immaterial to almost-sure convergence, because \(C^c\) has probability zero. Setting the sequence to zero there is simply a convenient way to define a measurable limit everywhere.
Worked Example: Partial Sums of a Bounded Random Series
Let \(B_1,B_2,\ldots\) be random variables taking values in \(\{0,1\}\), with no independence assumption, and define
For any outcome and any \(j,k\geq N\), suppose first that \(j\geq k\). Then
If \(k\geq j\), the same bound follows by interchanging \(j\) and \(k\). Given \(m\), choose \(N\) so that \(2^{-N}<1/m\). Every pair of terms from index \(N\) onward then differs by less than \(1/m\), for every outcome. Therefore \(P(C)=1\), and the theorem gives almost-sure convergence. In fact, the limit is the convergent series \(\sum_{r=1}^{\infty}B_r/2^r\), which lies in \([0,1]\) because \(0\leq B_r\leq1\).
Worked Example: Cauchy Paths Only on a Proper Event
Take \(\Omega=[0,1]\) with the uniform probability measure, let \(A=[0,1/3]\), and define
If \(\omega\in A\), then \(X_n(\omega)=1\) for every \(n\), so this path is Cauchy. If \(\omega\in A^c\), then \(X_n(\omega)=(-1)^n\), which alternates between \(1\) and \(-1\). For every \(N\), there are indices \(j,k\geq N\) of opposite parity, and for those indices \(|X_j(\omega)-X_k(\omega)|=2\). In particular, the Cauchy condition fails at tolerance \(1/2\). Thus \(C=A\) and \(P(C)=1/3\). The sequence is not Cauchy almost surely and, by the theorem, does not converge almost surely.
Why the Countable Formulation Matters
The exact equivalence relies on two facts: the real numbers are complete, and the sample-path Cauchy property can be expressed as a measurable event. The rational tolerances \(1/m\) handle both the quantifiers in the Cauchy condition and the measurability requirement. Using every positive real tolerance is mathematically natural, but it would write the event as an uncountable intersection, which need not be measurable merely because each individual event is measurable.
A common mistake is to check that the terms are close for each fixed pair of indices, or for a few fixed tolerances, and then conclude that the sequence is Cauchy. The condition is stronger: for each tolerance, there must be one index \(N\) after which all pairs of terms are close. The order of the quantifiers matters: the required \(N\) may depend on the tolerance and on the sample path, but it must work simultaneously for every \(j,k\geq N\).
Check Your Understanding
Use the sample-path Cauchy condition and the theorem to answer these questions.
- Why do the tolerances \(1/m\), for positive integers \(m\), suffice to test the Cauchy condition for every \(\varepsilon>0\)?
- Explain why the event \(\{|X_j-X_k|<1/m\}\) is measurable for fixed \(j,k,m\).
- If \(P(C)=1\), why does completeness of \(\mathbb R\) imply that every path in \(C\) has a finite limit?
- In the partial-sums example, why is independence of the \(B_r\) unnecessary?
- In the alternating-path example, which rational tolerance shows that paths in \(A^c\) are not Cauchy?