The Epsilon-Probability Test
A sequence can take values far from a proposed limit on some outcomes and still converge in probability, provided those outcomes become sufficiently unlikely. The test is made separately at each fixed positive tolerance: calculate the probability that the error exceeds that tolerance, then ask whether this probability tends to zero as the sequence index grows. The order of these quantifiers matters. The tolerance is fixed first; it is not allowed to depend on the index.
Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and let \(X_n\), \(n\ge 0\), and \(X\) be real-valued random variables on this space. The event \(\{|X_n-X|>\epsilon\}\) is the tail event that the absolute error exceeds the positive threshold \(\epsilon\). Its probability measures how often \(X_n\) lies more than \(\epsilon\) away from \(X\).
The equivalent form follows from the definition of a real sequence tending to zero: for each fixed \(\epsilon\), its tail probabilities must eventually be smaller than any prescribed \(\delta>0\). In applying the test, do not choose one convenient \(\epsilon\) and stop. Convergence requires the tail probability to vanish for every fixed positive \(\epsilon\). Conversely, a single fixed threshold for which the probabilities fail to vanish proves nonconvergence to the proposed limit.
Useful Forms of the Test
Proof. If \(X_n\xrightarrow{\mathbb{P}}X\), then the defining condition applies to each fixed threshold \(\epsilon=1/m>0\), giving the stated limits. For the converse, assume the limits hold for every positive integer \(m\), and take an arbitrary \(\epsilon>0\). Choose a positive integer \(m\) large enough that \(1/m<\epsilon\). Then \(\{|X_n-X|>\epsilon\}\subseteq\{|X_n-X|>1/m\}\), so \(\mathbb{P}(|X_n-X|>\epsilon)\le\mathbb{P}(|X_n-X|>1/m)\to0\). This holds for every \(\epsilon>0\), which is exactly convergence in probability. \(\square\)
This result can reduce an uncountable collection of threshold checks to a countable one. It does not say that one threshold is enough: the limits must still hold for every \(1/m\).
Proof. Suppose first that \(X_n\xrightarrow{\mathbb{P}}X\). For each \(\epsilon>0\), the event \(\{|X_n-X|\ge\epsilon\}\) is contained in \(\{|X_n-X|>\epsilon/2\}\). Therefore its probability tends to zero by the definition. In the other direction, assume the non-strict-threshold probabilities tend to zero for every positive threshold. Since \(\{|X_n-X|>\epsilon\}\subseteq\{|X_n-X|\ge\epsilon\}\), its probability also tends to zero for every \(\epsilon>0\). This is the defining condition. \(\square\)
Although either version can test convergence, the event in the definition uses a strict inequality. At equality, a value exactly \(\epsilon\) away from the limit does not belong to \(\{|X_n-X|>\epsilon\}\).
Worked Tail-Probability Calculations
Worked Example: A Deterministic Sequence
On any probability space, let \(X_n=1/n\) for \(n\ge1\), and let \(X=0\). Determine whether \(X_n\) converges in probability to \(X\).
Fix an arbitrary \(\epsilon>0\). Since \(X_n\) is deterministic, the event \(\{|X_n-X|>\epsilon\}\) is either the whole space or the empty set. Here it is the whole space exactly when \(1/n>\epsilon\), and it is empty when \(1/n\le\epsilon\). Thus
Choose \(N=\lfloor 1/\epsilon\rfloor+1\). For every \(n\ge N\), \(n>1/\epsilon\), so \(1/n<\epsilon\) and the tail probability is \(0\). Since this works for every \(\epsilon>0\), \(X_n\xrightarrow{\mathbb{P}}0\).
Check: Substituting \(X_n=1/n\) and \(X=0\) gives \(|X_n-X|=1/n\), the same threshold condition used above. If \(\epsilon=1/4\), then \(N=\lfloor4\rfloor+1=5\); for all \(n\ge5\), \(1/n\le1/5<1/4\), so the probability is indeed zero.
Worked Example: Bernoulli Variables with Vanishing Success Probability
Take \(\Omega=[0,1]\) with its Borel sets and uniform probability. Define \(X_n=\mathbf{1}_{[0,\,1/(n+2)]}\), so \(X_n\) is Bernoulli with success probability \(1/(n+2)\). Test convergence in probability to \(X=0\).
For \(0<\epsilon<1\), the error is \(1\) on \([0,1/(n+2)]\) and \(0\) elsewhere. The value \(1\) exceeds \(\epsilon\), so \(\mathbb{P}(|X_n|>\epsilon)=1/(n+2)\). For \(\epsilon\ge1\), neither possible value of \(|X_n|\), namely \(0\) and \(1\), is strictly greater than \(\epsilon\). Thus the probability is \(0\), including at \(\epsilon=1\). In both cases the tail probability tends to zero, so \(X_n\xrightarrow{\mathbb{P}}0\).
Check: At the specific threshold \(\epsilon=1/2\), the event is exactly the success event, and its probability is \(1/(n+2)\to0\). At \(\epsilon=1\), the event is empty because \(1>1\) is false; its probability is \(0\).
Worked Example: A Scaled Random Variable
On \(\Omega=[0,1]\) with uniform probability, let \(Y=2\mathbf{1}_{[0,\,1/2]}\), and define \(X_n=Y/n\) and \(X=0\) for \(n\ge1\). The variable \(Y\) equals \(2\) with probability \(1/2\) and \(0\) with probability \(1/2\). Decide whether \(X_n\) converges in probability to zero.
For a fixed \(\epsilon>0\), the error \(|X_n|\) is \(2/n\) with probability \(1/2\) and \(0\) with probability \(1/2\). Therefore
Choose \(N=\lfloor2/\epsilon\rfloor+1\). Then \(n\ge N\) implies \(n>2/\epsilon\), hence \(2/n<\epsilon\), and the tail probability is zero. This holds for each fixed positive \(\epsilon\), so \(X_n\xrightarrow{\mathbb{P}}0\).
Check: At \(\epsilon=1/3\), \(N=\lfloor6\rfloor+1=7\). For \(n=7\), the nonzero error is \(2/7<1/3\), so the event is empty; for every \(n\ge7\), it remains empty.
Worked Example: A Bernoulli Sequence That Fails the Test
On \([0,1]\) with uniform probability, let \(B=\mathbf{1}_{[0,\,1/2]}\), and set \(X_n=B\) for every \(n\). Test convergence in probability to zero.
At \(\epsilon=1/2\), the error is \(1\) on \([0,1/2]\) and \(0\) on the rest of the space. Since \(1>1/2\), \(\mathbb{P}(|X_n|>1/2)=1/2\) for every \(n\). This probability does not tend to zero. Therefore \(X_n\) does not converge in probability to zero.
Check: The same calculation works for every fixed \(0<\epsilon<1\): the tail probability is \(1/2\). In particular, the failure is not caused by choosing a threshold at which the error equals the threshold.
How to Use the Test Reliably
A tail calculation usually has three parts: identify the possible values of the error, determine exactly which values satisfy the strict inequality, and sum their probabilities. For a discrete random variable, this is a finite or countable probability calculation. For deterministic variables, the event probability is either zero or one. For a scaled variable, first compute the scale of the error and then compare it with the fixed threshold.
For instance, if a Bernoulli variable takes values \(0\) and \(1\), then for a target of zero and \(0<\epsilon<1\), its tail event is precisely the success event. This makes its success probability the relevant quantity. But at \(\epsilon=1\), the strict tail event is empty. Both facts follow directly from the inequality defining the event, not from an informal notion of being “close.”
A common error is to replace \(>\epsilon\) by \(\ge\epsilon\) without noticing the change. For the deterministic sequence \(X_n=1/n\), at \(\epsilon=1/4\) and \(n=4\), the error is exactly \(1/4\). The correct strict event \(\{|X_4|>1/4\}\) is empty, so its probability is \(0\). The incorrect calculation \(\mathbb{P}(|X_4|\ge1/4)=1\) includes equality. The non-strict version is also a valid convergence test, as proved above, but it must be applied consistently; it is not the same event at each \(n\) and threshold.
Another mistake is checking only one value of \(\epsilon\). The failure example shows how one threshold can disprove convergence, but passing one threshold does not prove it. The definition requires every fixed positive threshold. The countable-threshold result gives a systematic alternative: check \(1,1/2,1/3,\ldots\), not just one of these values.
Finally, the definition concerns the probability of an error event, not whether the error is small on every outcome. A tail probability can tend to zero even when each \(X_n\) has a nonzero chance of differing from \(X\); the Bernoulli example with success probability \(1/(n+2)\) illustrates this. The relevant question at each fixed \(\epsilon\) is whether the chance of an error larger than \(\epsilon\) vanishes.
Check Your Understanding
For each question, write down the strict tail event before evaluating its probability.
- If \(X_n=2/n\) deterministically and \(X=0\), compute \(\mathbb{P}(|X_n-X|>1/5)\) for \(n=9\) and \(n=10\). Does the sequence converge in probability to zero?
- Let \(X_n\) be Bernoulli with success probability \(1/(n+1)\). For which positive thresholds is \(\mathbb{P}(|X_n|>\epsilon)\) equal to \(1/(n+1)\), and for which is it zero?
- For the variable \(Y\) in the scaled example, compute \(\mathbb{P}(|Y/4|>1/2)\). Explain the role of strict inequality.
- Give one fixed threshold that proves the constant Bernoulli sequence in the failure example does not converge in probability to zero.
- Why does checking the thresholds \(1/m\) for all positive integers \(m\) suffice to establish convergence in probability?