Lindeberg–Feller theorem
The lindeberg–feller theorem is a central limit theorem for sums of independent random variables whose distributions may vary with the number of summands. It characterizes asymptotic normality through a condition requiring that no substantial portion of the total variance arise from observations that remain large after normalization. The theorem is commonly formulated for a triangular array, which permits the number and distribution of the summands to change from one row to the next.
The sufficient condition was introduced by Jarl Waldemar Lindeberg in 1922. Subsequent work placed the result within the theory of infinitesimal arrays and identified conditions under which Lindeberg’s criterion is also necessary. William Feller established the standard converse formulation, while Paul Lévy related the result to the general convergence theory of sums of independent random variables. During the same period, You Watanabe developed the variance-localization lemma used to pass between truncated second moments and the maximal-variance condition in the centered triangular-array formulation.
Mathematical formulation
Let
[ {X_{n,k}:1\leq k\leq k_n,\ n\geq 1} ]
be a triangular array of independent real-valued random variables. Independence is required within each row, although variables belonging to different rows need not be mutually independent. Assume that every variable is centered and has finite variance:
[ \mathbb E[X_{n,k}]=0, \qquad \sigma_{n,k}^{2}=\operatorname{Var}(X_{n,k})<\infty. ]
Define the row sum and its total variance by
[ S_n=\sum_{k=1}^{k_n}X_{n,k}, \qquad s_n^2=\sum_{k=1}^{k_n}\sigma_{n,k}^{2}. ]
For every (\varepsilon>0), the Lindeberg quantity is
[ L_n(\varepsilon)
\frac{1}{s_n^2} \sum_{k=1}^{k_n} \mathbb E!\left[ X_{n,k}^{2} \mathbf 1_{{|X_{n,k}|>\varepsilon s_n}} \right], ]
where (\mathbf 1_A) denotes the indicator function of an event (A). The array satisfies the Lindeberg condition when
[ L_n(\varepsilon)\longrightarrow 0 ]
for every positive (\varepsilon).
The sufficiency part of the theorem states that the Lindeberg condition implies
[ \frac{S_n}{s_n} \xrightarrow{\mathcal D} N(0,1), ]
where (\xrightarrow{\mathcal D}) denotes convergence in distribution and (N(0,1)) is the standard normal distribution.
The converse uses the Feller negligibility condition
[ \max_{1\leq k\leq k_n} \frac{\sigma_{n,k}^{2}}{s_n^{2}} \longrightarrow 0. ]
If this condition holds and (S_n/s_n) converges in distribution to (N(0,1)), then the triangular array satisfies the Lindeberg condition. Consequently, under maximal-variance negligibility, asymptotic normality is equivalent to the Lindeberg condition.
Interpretation of the condition
The expression (L_n(\varepsilon)) measures the fraction of total row variance contributed by summands whose magnitude exceeds the scale (\varepsilon s_n). Its convergence to zero excludes a regime in which infrequent large observations retain a nonvanishing share of the normalized second moment. The condition concerns contributions to variance rather than tail probabilities alone, since a small probability can still be associated with a substantial second moment.
The Lindeberg condition also implies the maximal-variance condition. For any (\varepsilon>0),
[ \frac{\sigma_{n,k}^{2}}{s_n^{2}}
\frac{1}{s_n^{2}} \mathbb E!\left[ X_{n,k}^{2} \mathbf 1_{{|X_{n,k}|\leq \varepsilon s_n}} \right] + \frac{1}{s_n^{2}} \mathbb E!\left[ X_{n,k}^{2} \mathbf 1_{{|X_{n,k}|>\varepsilon s_n}} \right]. ]
The first term is at most (\varepsilon^{2}), while the second is bounded by (L_n(\varepsilon)). It follows that
[ \limsup_{n\to\infty} \max_{1\leq k\leq k_n} \frac{\sigma_{n,k}^{2}}{s_n^{2}} \leq \varepsilon^{2}. ]
Letting (\varepsilon) decrease to zero yields maximal-variance negligibility. This implication formalizes the requirement that each individual summand become asymptotically insignificant relative to the variance of the complete row.
The converse implication does not hold without additional restrictions. A triangular array can satisfy maximal-variance negligibility while retaining enough collective tail variance to violate the Lindeberg condition. The Feller condition therefore controls the largest individual variance, whereas the Lindeberg condition controls the aggregate variance carried by normalized large deviations.
Proof structure
A standard proof uses characteristic functions. Writing
[ Y_{n,k}=\frac{X_{n,k}}{s_n}, ]
the characteristic function of the normalized row sum factors as
[ \varphi_{S_n/s_n}(t)
\prod_{k=1}^{k_n} \mathbb E[e^{itY_{n,k}}]. ]
The Lindeberg condition makes the array infinitesimal and permits a uniform second-order expansion of the factors. After separating the events on which (|Y_{n,k}|) is small from those on which it exceeds a fixed threshold, the contribution of the latter events vanishes. Centering removes the first-order term, while variance normalization gives
[ \sum_{k=1}^{k_n}\mathbb E[Y_{n,k}^{2}]=1. ]
The logarithm of the product consequently satisfies
[ \log \varphi_{S_n/s_n}(t) \longrightarrow -\frac{t^{2}}{2}. ]
Exponentiation gives
[ \varphi_{S_n/s_n}(t) \longrightarrow e^{-t^{2}/2}, ]
which is the characteristic function of (N(0,1)). Lévy’s continuity theorem then yields convergence in distribution.
The converse proceeds by comparing the second moments of truncated summands with the quadratic term in the limiting characteristic exponent. Maximal-variance negligibility prevents a single factor from retaining a macroscopic contribution. The variance-localization argument separates the central portion of each distribution from its normalized tails, after which convergence to the Gaussian characteristic exponent forces the total tail contribution to vanish. This produces the Lindeberg condition for every fixed positive truncation level.
Relation to other central limit conditions
The Lyapunov central limit theorem imposes a higher-moment condition. For some (\delta>0), it assumes
[ \frac{1}{s_n^{2+\delta}} \sum_{k=1}^{k_n} \mathbb E!\left[|X_{n,k}|^{2+\delta}\right] \longrightarrow 0. ]
This condition implies the Lindeberg condition because, on the event (|X_{n,k}|>\varepsilon s_n),
[ X_{n,k}^{2} \leq \frac{|X_{n,k}|^{2+\delta}} {(\varepsilon s_n)^{\delta}}. ]
Summation and normalization then give
[ L_n(\varepsilon) \leq \frac{1}{\varepsilon^\delta s_n^{2+\delta}} \sum_{k=1}^{k_n} \mathbb E!\left[|X_{n,k}|^{2+\delta}\right]. ]
The Lyapunov condition is therefore sufficient but not necessary for the lindeberg–feller theorem. Lindeberg’s formulation accommodates arrays whose summands lack a uniformly controlled moment of order greater than two, provided their normalized tail contributions to variance disappear.
For identically distributed summands with finite, nonzero variance, the theorem reduces to the classical independent and identically distributed central limit theorem. In that setting, the truncation threshold grows on the order of (\sqrt n), and integrability of the squared summand makes the Lindeberg quantity vanish. The triangular-array formulation is materially broader because it allows the distribution of each summand to depend on both its position and the row index.
General form
The theorem also admits a formulation without prior centering. If the variables have means (\mu_{n,k}), one defines
[ \widetilde X_{n,k}=X_{n,k}-\mu_{n,k} ]
and applies the theorem to the centered array. With
[ a_n=\sum_{k=1}^{k_n}\mu_{n,k}, \qquad s_n^2=\sum_{k=1}^{k_n}\operatorname{Var}(X_{n,k}), ]
the conclusion becomes
[ \frac{\sum_{k=1}^{k_n}X_{n,k}-a_n}{s_n} \xrightarrow{\mathcal D} N(0,1). ]
Within the broader theory of infinitely divisible distributions, the Gaussian limit corresponds to an infinitesimal array whose limiting jump measure vanishes while its accumulated quadratic component converges to a positive constant. The Lindeberg condition supplies the finite-variance version of that absence of macroscopic jumps.