Infinitesimal Arrays and Feller's Condition

Why the Largest Component Matters

A central limit theorem for a sum of independent variables is most credible when no single summand determines the sum’s behavior. But “no single summand matters” can refer to different things: a summand may be unlikely to be large, or its variance may be small relative to the total variance. These are related ideas, but they are not interchangeable.

For a triangular array, the number and distribution of variables can change with the row. We will compare each component with the scale of the whole row, then identify precisely what infinitesimality and Feller’s condition say. Their role is clearest alongside the Lindeberg condition: Lindeberg controls the total contribution of large components, while Feller’s condition limits the variance carried by the largest one.

Definitions and Normalization

For each $n$, let $X_{n1},\ldots,X_{n k_n}$ be independent random variables with $E[X_{nk}]=0$ and finite variances. Define the total row variance and its positive square root by

s_n^2=\sum_{k=1}^{k_n}\operatorname{Var}(X_{nk}), \qquad s_n>0.

The normalized row sum is $W_n=s_n^{-1}\sum_{k=1}^{k_n}X_{nk}$. To compare individual components on the same scale as this sum, write $Y_{nk}=X_{nk}/s_n$. Then $\sum_k\operatorname{Var}(Y_{nk})=1$. Statements about component size for a normal-limit problem are usually made on this normalized scale.

Definition: The array $\{Y_{nk}\}$ is infinitesimal if, for every $\varepsilon>0$, $$ \max_{1\leq k\leq k_n}P(|Y_{nk}|>\varepsilon)\longrightarrow 0. $$ In words, the probability that any particular component exceeds a fixed positive threshold must become uniformly small across the row. For the original variables $X_{nk}$, the corresponding normalized definition is $\max_k P(|X_{nk}|>\varepsilon s_n)\to0$.

Definition: Feller’s condition for the array is $$ \frac{\max_{1\leq k\leq k_n}\operatorname{Var}(X_{nk})}{s_n^2}\longrightarrow 0. $$ Equivalently, $\max_k\operatorname{Var}(Y_{nk})\to0$: the largest component variance, measured as a fraction of total row variance, vanishes.

Feller’s condition concerns variances, not probabilities of large values. Infinitesimality concerns probabilities at fixed thresholds, not the full variance contribution. The distinction matters for variables with rare but very large values.

How Lindeberg Controls the Largest Component

For each fixed $\varepsilon>0$, the Lindeberg quantity is

L_n(\varepsilon)=\frac{1}{s_n^2}\sum_{k=1}^{k_n} E\!\left[X_{nk}^2\mathbf{1}_{\{|X_{nk}|>\varepsilon s_n\}}\right].

The Lindeberg condition is $L_n(\varepsilon)\to0$ for every $\varepsilon>0$. It requires the aggregate variance contribution from components exceeding a fixed fraction of the total standard deviation to vanish.

Theorem: If the centered array satisfies the Lindeberg condition, then it satisfies Feller’s condition and is infinitesimal on the normalized scale.

The two conclusions follow from separate inequalities. First, fix $\varepsilon>0$ and any component $k$. Pointwise,

\mathbf{1}_{\{|X_{nk}|>\varepsilon s_n\}} \leq \frac{X_{nk}^2}{\varepsilon^2s_n^2} \mathbf{1}_{\{|X_{nk}|>\varepsilon s_n\}}.

On the event in the indicator, $X_{nk}^2/(\varepsilon^2s_n^2)>1$; off that event both sides are zero. Taking expectations therefore gives the tail-second-moment bound

P(|X_{nk}|>\varepsilon s_n) \leq \frac{E[X_{nk}^2\mathbf{1}_{\{|X_{nk}|>\varepsilon s_n\}}]}{\varepsilon^2s_n^2}.

Taking the maximum over $k$, and bounding a single nonnegative summand by the sum of all such summands, yields

\max_k P(|X_{nk}|>\varepsilon s_n) \leq \frac{L_n(\varepsilon)}{\varepsilon^2}\longrightarrow0.

This proves normalized infinitesimality.

For Feller’s condition, split each component’s second moment at the same threshold. Since $E[X_{nk}]=0$, its variance is $E[X_{nk}^2]$, and

\begin{aligned} \frac{\operatorname{Var}(X_{nk})}{s_n^2} &=\frac{E[X_{nk}^2\mathbf{1}_{\{|X_{nk}|\leq\varepsilon s_n\}}]}{s_n^2} +\frac{E[X_{nk}^2\mathbf{1}_{\{|X_{nk}|>\varepsilon s_n\}}]}{s_n^2}\\ &\leq \varepsilon^2+L_n(\varepsilon). \end{aligned}

The first term is at most $\varepsilon^2$, because $X_{nk}^2\leq\varepsilon^2s_n^2$ on the event being integrated. The second is at most the whole-row Lindeberg quantity. Taking the maximum over $k$ and then the limit superior gives

\limsup_{n\to\infty}\max_k\frac{\operatorname{Var}(X_{nk})}{s_n^2} \leq\varepsilon^2.

This holds for every $\varepsilon>0$. Letting $\varepsilon$ decrease to zero proves Feller’s condition. $\square$

Feller’s Condition Also Implies Infinitesimality

For centered components, Feller’s condition itself implies normalized infinitesimality. Indeed, Chebyshev’s inequality gives, for each $\varepsilon>0$,

\max_k P(|Y_{nk}|>\varepsilon) \leq \frac{\max_k\operatorname{Var}(Y_{nk})}{\varepsilon^2} \longrightarrow0.

Thus, in this finite-variance centered setting, Feller’s condition is a stronger way to ensure that no individual component has appreciable variance, and it guarantees infinitesimality. The converse need not hold: probabilities can become small even while rare, extreme values preserve a substantial component variance.

Worked Example: Equal-Variance Normal Components

Let $X_{n1},\ldots,X_{nn}$ be independent $N(0,1)$ variables. Here $k_n=n$, and the total variance is $s_n^2=n$.

Each component’s fraction of total variance is

\frac{\operatorname{Var}(X_{nk})}{s_n^2}=\frac{1}{n}, \qquad \frac{\max_k\operatorname{Var}(X_{nk})}{s_n^2}=\frac{1}{n}\longrightarrow0.

Feller’s condition holds. The normalized components are $Y_{nk}=X_{nk}/\sqrt{n}$, so $Y_{nk}\sim N(0,1/n)$. For any fixed $\varepsilon>0$, Chebyshev’s inequality gives

\max_k P(|Y_{nk}|>\varepsilon)\leq\frac{1}{n\varepsilon^2}\longrightarrow0.

The normalized array is infinitesimal as well. This example illustrates a common setting in which each term contributes a vanishing fraction of the total variance.

Worked Example: Infinitesimal but Not Feller

For each $n$, let the row contain just one centered random variable $X_{n1}$ with distribution

P(X_{n1}=n)=\frac{1}{2n^2},\qquad P(X_{n1}=-n)=\frac{1}{2n^2},\qquad P(X_{n1}=0)=1-\frac{1}{n^2}.

Symmetry gives $E[X_{n1}]=0$, and its variance is

s_n^2=\operatorname{Var}(X_{n1})=E[X_{n1}^2] =n^2\left(\frac{1}{n^2}\right)=1.

For any fixed $\varepsilon>0$, once $n>\varepsilon$,

P(|X_{n1}|>\varepsilon s_n)=\frac{1}{n^2}\longrightarrow0.

So the normalized array is infinitesimal. But it fails Feller’s condition: the only component carries all the row variance, and $\max_k\operatorname{Var}(X_{nk})/s_n^2=1$ for every $n$. It also fails Lindeberg’s condition, since for $n>\varepsilon$,

L_n(\varepsilon)=E[X_{n1}^2\mathbf{1}_{\{|X_{n1}|>\varepsilon\}}]=1.

Small exceedance probabilities alone do not rule out substantial variance from very rare extremes.

Worked Example: Feller Holds but Lindeberg Fails

For each $n$, take $n$ independent, identically distributed variables in row $n$, each with distribution

P(X_{nk}=\sqrt n)=\frac{1}{2n},\qquad P(X_{nk}=-\sqrt n)=\frac{1}{2n},\qquad P(X_{nk}=0)=1-\frac{1}{n}.

Each component is centered and has variance $n(1/n)=1$, so $s_n^2=n$. The largest variance fraction is $1/n\to0$, and Feller’s condition holds. For fixed $\varepsilon>0$ and $n>\varepsilon^2$, the normalized component exceeds $\varepsilon$ in absolute value exactly when it is nonzero. Hence

\max_k P(|X_{nk}|>\varepsilon s_n)=\frac{1}{n}\longrightarrow0.

The array is infinitesimal. Yet, for the same $n>\varepsilon^2$, every component’s squared value $n$ exceeds the threshold $\varepsilon^2s_n^2=\varepsilon^2n$ whenever it is nonzero. Consequently,

L_n(\varepsilon) =\frac{1}{n}\sum_{k=1}^{n}E[X_{nk}^2\mathbf{1}_{\{|X_{nk}|>\varepsilon\sqrt n\}}] =\frac{1}{n}\cdot n\cdot 1 =1.

Thus Feller’s condition does not imply the Lindeberg condition. Each component has a small variance share, but the row’s variance is entirely in values above the fixed normalized threshold.

Role in Normal-Limit Criteria

The Lindeberg–Feller central limit theorem provides the main connection: for independent, centered triangular arrays with $s_n^2>0$, the Lindeberg condition implies

\frac{1}{s_n}\sum_{k=1}^{k_n}X_{nk}\xrightarrow{d}N(0,1).

The preceding result shows that Lindeberg’s condition automatically rules out a variance-dominating component and ensures infinitesimality. Feller’s condition is therefore a useful diagnostic and a necessary companion in many formulations of normal-limit criteria. But it does not replace Lindeberg’s tail control: the third example has Feller’s condition and infinitesimality but fails Lindeberg.

Keep the three statements distinct:

Infinitesimality: each component’s probability of exceeding a fixed normalized threshold vanishes uniformly.
Feller’s condition: the largest component variance is negligible relative to total variance.
Lindeberg’s condition: the sum of normalized second-moment contributions beyond every fixed threshold vanishes.

Common Pitfall

A tempting but incorrect argument is: “The probability of a large component is small, so its variance must be small.” Variance weights an outcome by its squared magnitude. A sufficiently extreme value can contribute a fixed amount to variance even when its probability tends to zero, as the infinitesimal but non-Feller example shows.

When bounding a tail probability by a tail second moment, the indicator must remain on the right-hand side. The valid pointwise inequality is

\mathbf{1}_{\{|X|>a\}}\leq\frac{X^2}{a^2}\mathbf{1}_{\{|X|>a\}}, \qquad a>0,

not merely an inequality that produces the untruncated second moment. This distinction is essential when the argument relies on a condition specifically controlling tail second moments.

For a normal-limit problem, check what is actually assumed: infinitesimality rules out frequent large terms; Feller’s condition rules out a dominant variance share; and Lindeberg controls the aggregate variance in the tails. They answer different questions, even when one condition implies another.

Check Your Understanding

Try these questions before moving on.

State Feller’s condition in terms of the normalized components $Y_{nk}$, and explain what it says about the largest variance share.
Why does Feller’s condition imply normalized infinitesimality for centered components with finite variances?
In the rare-extreme example with one component, which condition holds: infinitesimality, Feller’s condition, or Lindeberg’s condition?
What does the Lindeberg condition control that Feller’s condition does not?
For $n$ independent row components, each with variance $1$, calculate the largest variance fraction when the total row variance is $n$.