Ratios With a Vanishing Limiting Denominator

Why a Vanishing Denominator Changes the Problem

A ratio can be unstable even when both of its parts are small. If a denominator approaches a nonzero constant, Slutsky's theorem gives a direct rule for the ratio. If it approaches zero, that rule no longer applies: small changes in the denominator can have a large effect, and the ratio depends on how quickly numerator and denominator shrink together.

The key distinction is between knowing that \(A_n\) and \(B_n\) each approach zero and knowing their joint behavior at suitable rates. The first fact alone says little about \(A_n/B_n\). For a limit theorem, we need scaled numerator and denominator limits and information about their joint distribution.

Definition: Let \(A_n\) and \(B_n\) be random variables, and let \(c_n,d_n\) be nonzero deterministic scaling sequences. A rate-scaled pair is \((c_nA_n,d_nB_n)\). Its joint distributional limit describes the numerator and denominator on the scales relevant to their ratio. If \(A_n\) and \(B_n\) both converge in probability to zero, the denominator is said here to have a vanishing limiting value; this fact by itself does not specify the rate-scaled pair.

For comparison, the ordinary ratio rule applies if \(A_n\xrightarrow{d}A\) and \(B_n\xrightarrow{p}b\) for a constant \(b\ne 0\): then \(A_n/B_n\xrightarrow{d}A/b\), provided the ratios are defined. When \(b=0\), division is not a continuous operation at the limiting pair \((A,0)\), and this rule cannot be invoked. In particular, convergence of a denominator to zero is not a substitute for convergence to a nonzero constant.

A Ratio Limit After Rescaling

The useful approach is to find a scale for each component separately. The algebra fixes which scale must then be applied to the ratio:

\frac{c_nA_n}{d_nB_n} = \frac{c_n}{d_n}\frac{A_n}{B_n}.

Thus the factor multiplying the unscaled ratio is \(c_n/d_n\), not its reciprocal. The next result states conditions under which this scaled ratio has a limit.

Theorem: Suppose \(c_n\) and \(d_n\) are nonzero deterministic numbers and \[ (c_nA_n,d_nB_n)\xrightarrow{d}(X,Y), \qquad P(Y=0)=0. \] Define the ratio arbitrarily on any event where \(B_n=0\). Then \[ \frac{c_n}{d_n}\frac{A_n}{B_n}\xrightarrow{d}\frac{X}{Y}. \] In particular, the probability that \(B_n=0\) tends to zero, so the arbitrary definition on that event does not affect the limit.

Proof. By the continuous mapping theorem, the map \(g(x,y)=x/y\), which is continuous at every point with \(y\ne0\), gives \[ \frac{c_nA_n}{d_nB_n}\xrightarrow{d}\frac{X}{Y} \] when the ratio is defined wherever its denominator is nonzero. To check that zero denominators become negligible, use \(d_n\ne0\) and the closed-set part of the Portmanteau theorem:

\limsup_{n\to\infty}P(B_n=0) =\limsup_{n\to\infty}P(d_nB_n=0) \le P(Y=0)=0.

The chosen value of a ratio on \(\{B_n=0\}\) can therefore affect it only on an event whose probability tends to zero. Replacing random variables on events with probability tending to zero does not change their distributional limit. Finally, the displayed algebra gives \((c_nA_n)/(d_nB_n)=(c_n/d_n)(A_n/B_n)\) wherever \(B_n\ne0\), proving the result. \(\square\)

Worked Rate-Scaled Examples

Worked Example: Equal Rates Produce a Cauchy Ratio

Let \(Z\) and \(W\) be independent \(N(0,1)\) random variables, and set \(A_n=Z/\sqrt n\) and \(B_n=W/\sqrt n\). Both components converge in probability to zero. Choose \(c_n=d_n=\sqrt n\). Then \((c_nA_n,d_nB_n)=(Z,W)\) for every \(n\), so the joint limit is \((Z,W)\). Since \(P(W=0)=0\), the theorem gives \[ \frac{c_n}{d_n}\frac{A_n}{B_n}=\frac{A_n}{B_n}\xrightarrow{d}\frac{Z}{W}. \]

To identify this limit, transform from \((Z,W)\) to \((R,W)=(Z/W,W)\). For \(w\ne0\), the inverse map is \((r,w)\mapsto(rw,w)\), whose Jacobian has absolute value \(|w|\). The density of \(R\) is therefore \[ \begin{aligned} f_R(r) &=\int_{-\infty}^{\infty}\frac{1}{2\pi} \exp\!\left(-\frac{(r^2+1)w^2}{2}\right)|w|\,dw\\ &=\frac{1}{\pi(1+r^2)}. \end{aligned} \] For the last equality, the integral of \(|w|\exp(-aw^2/2)\) over the real line is \(2/a\), with \(a=1+r^2\). This is the standard Cauchy density. In this example the ratio has that distribution for every \(n\), not just in the limit.

Worked Example: Unequal Rates and the Ratio Scaling Factor

Using independent \(Z,W\sim N(0,1)\), let \(A_n=Z/n\) and \(B_n=W/\sqrt n\). Here \(A_n\) shrinks faster than \(B_n\). Set \(c_n=n\) and \(d_n=\sqrt n\); then \(c_nA_n=Z\), \(d_nB_n=W\), and the joint scaled limit is again \((Z,W)\), with \(P(W=0)=0\).

The unscaled ratio and the required factor are \[ \frac{A_n}{B_n}=\frac{Z}{\sqrt n\,W}, \qquad \frac{c_n}{d_n}=\sqrt n. \] Consequently, \[ \frac{c_n}{d_n}\frac{A_n}{B_n} =\sqrt n\frac{Z}{\sqrt n\,W} =\frac{Z}{W}\xrightarrow{d}\text{standard Cauchy}. \] The factor is \(c_n/d_n\) because it reconstructs the ratio of the scaled variables. Using \(d_n/c_n=1/\sqrt n\) instead would give \(Z/(nW)\), which converges in probability to zero rather than producing the nondegenerate Cauchy limit.

Worked Example: Marginal Limits Do Not Determine the Ratio

Consider first \(A_n=Z/\sqrt n\) and \(B_n=Z/\sqrt n\), with \(Z\sim N(0,1)\). Each scaled component is exactly \(N(0,1)\), and both unscaled components converge in probability to zero. But \(A_n/B_n=1\) whenever \(Z\ne0\), an event of probability one.

Now let \(A_n=Z/\sqrt n\) and \(B_n=W/\sqrt n\), with independent standard normal \(Z,W\). The marginal distributions and componentwise rates are unchanged, but the ratio has a standard Cauchy distribution. In both examples the joint scaled pair has a limit and its second coordinate is nonzero with probability one. The ratio limits differ because the joint limits differ: one has perfectly dependent coordinates and the other has independent coordinates. Thus the two marginal limits, even with matching rates, do not determine the ratio limit.

What Information Is Missing?

If all that is known is \(A_n\xrightarrow{p}0\) and \(B_n\xrightarrow{p}0\), there need not be a ratio limit. For deterministic sequences, take \(A_n=1/n\). If \(B_n=1/n\), the ratio is 1; if \(B_n=1/n^2\), it is \(n\); if \(B_n=2/n\), it is \(1/2\). All these denominators tend to zero. The ratio can even fail to converge: set \(B_n=1/n\) for even \(n\) and \(B_n=2/n\) for odd \(n\). Then both sequences still tend to zero, while \(A_n/B_n\) alternates between 1 and \(1/2\).

A vanishing denominator also does not guarantee that a scaled ratio has a finite, nondegenerate limit. For instance, if \(A_n=1\) and \(B_n=1/n\), the ratio is \(n\), which escapes to infinity rather than converging to a real-valued distribution. In the rate-scaled theorem, the condition \(P(Y=0)=0\) is essential: if the limiting scaled denominator can equal zero, the ratio map is discontinuous on a set with positive limiting probability, and this theorem does not settle what happens there.

Why the Ordinary Ratio Delta Method Fails

For \(g(x,y)=x/y\), the gradient at a point with \(y\ne0\) is \[ \nabla g(x,y)=\left(\frac{1}{y},-\frac{x}{y^2}\right). \] A first-order delta-method approximation around a fixed point requires the function to be differentiable there. At \((0,0)\), \(g\) is not defined; near points with small nonzero \(y\), the gradient components can grow without bound. So one cannot simply use the ordinary ratio delta method with a limiting point whose denominator is zero.

Even when the denominator is nonzero at every finite \(n\), a sequence of expansion points approaching zero can make the linear approximation unreliable unless the fluctuation sizes and the Taylor remainder are controlled on the same scales. The rate-scaled theorem avoids assuming differentiability at \((0,0)\): it first obtains a joint limit on the relevant scales and then applies the continuous mapping theorem at the limiting pair \((X,Y)\), where \(Y\ne0\) almost surely.

Common Mistakes and a Practical Check

Applying Slutsky with a zero denominator limit. The usual ratio rule requires a nonzero denominator limit. When the limit is zero, identify additional rate information instead.
Using the reciprocal scaling factor. If the scaled components are \(c_nA_n\) and \(d_nB_n\), their ratio is \((c_n/d_n)(A_n/B_n)\). Check this identity before stating a limit.
Using marginal convergence alone. Ratios depend on the joint behavior, including dependence between numerator and denominator.
Ignoring a zero-denominator event. Ensure the ratio is defined, or show that \(P(B_n=0)\to0\). Under the theorem's assumptions, the latter follows from \(P(Y=0)=0\).

A useful diagnostic is to write down the proposed scales and verify each step: do the two scaled variables have a joint distributional limit? Is the limiting denominator nonzero almost surely? Does the proposed factor equal \(c_n/d_n\)? If any answer is missing, the ratio limit has not yet been justified.

Key takeaway: Convergence of numerator and denominator to zero does not determine a ratio limit. Find scales \(c_n,d_n\) yielding a joint limit \((X,Y)\), check \(P(Y=0)=0\), and then use \((c_n/d_n)(A_n/B_n)\xrightarrow{d}X/Y\).

Check Your Understanding

Use the definitions and conditions in the tutorial to answer each question.

Why does the ordinary ratio rule not apply when the denominator converges in probability to zero?
If \(c_nA_n\xrightarrow{d}X\) and \(d_nB_n\xrightarrow{d}Y\) jointly, which factor multiplies \(A_n/B_n\) to obtain the ratio of the scaled variables?
For \(A_n=Z/n\) and \(B_n=W/\sqrt n\), with independent standard normal \(Z,W\), calculate the scaled ratio using \(c_n=n\) and \(d_n=\sqrt n\).
Why are the marginal distributions of the rate-scaled numerator and denominator insufficient to determine the ratio limit?
What condition on \(Y\) permits applying the continuous mapping theorem to the ratio \(X/Y\)?