Chen–Stein Poisson approximation: Definition & Examples

Chen–Stein Poisson approximation is a method for bounding the error when a sum of rare-event indicators is replaced by a Poisson distribution. For a quant counting threshold breaches or operational exceptions, the useful output is an error bound on event probabilities. A matched expected count alone does not justify the approximation.

Write

W=\sum_{i=1}^n X_i

, where each

X_i

has a Bernoulli distribution with probability

p_i

, and set

\lambda=\mathbb{E}[W]=\sum_i p_i>0

. The comparison variable is

Z\sim\text{Poisson}(\lambda)

. Independence is one sufficient setting; dependent indicators need additional control.

What does the error bound mean?

Use the following convention for total variation distance:

d_{\mathrm{TV}}(W,Z) =\sup_{A\subseteq\{0,1,2,\ldots\}} \lvert\mathbb{P}(W\in A)-\mathbb{P}(Z\in A)\rvert =\frac12\sum_{k=0}^{\infty}\lvert\mathbb{P}(W=k)-\mathbb{P}(Z=k)\rvert.

A bound of

0.02

means every count-event probability differs by at most two percentage points. It is an absolute error guarantee, not a 2% relative error guarantee. The papers cited below also use a norm equal to the full sum, twice our

d_{\mathrm{TV}}

; their constants must be converted accordingly.

For independent indicators, the coarse Le Cam inequality is

d_{\mathrm{TV}}(W,Z)\le\sum_{i=1}^n p_i^2.

The

p_i

need not be equal. Small individual probabilities help, but their squares must be summed: “rare” is not a substitute for checking the bound. This independent result predates Chen's extension of Stein's method. Le Cam's Proposition 1 gives it in the full-sum norm.

Fifty opportunities, one expected event

Consider a synthetic one-day exception model: 50 distinct opportunities each produce either zero or one exception, independently, with probability

0.02

. The count has a binomial distribution,

W\sim\text{Binomial}(50,0.02)

, and

\lambda=50(0.02)=1

The exact and approximate probabilities of a quiet day are

\begin{aligned} \mathbb{P}(W=0)&=0.98^{50}\approx0.3641696801,\\ \mathbb{P}(Z=0)&=e^{-1}\approx0.3678794412. \end{aligned}

Their difference is about

0.0037097611

, or 0.371 percentage points. Meanwhile,

\sum_i p_i^2=50(0.02)^2=0.02.

Thus the error for the event “zero exceptions” satisfies the guarantee. It is only one event: the actual total variation is about $0.0055834098$ , attained by the set of counts $\{1,2\}$ . Neither number equals the upper bound.

Synthetic Binomial(50, 0.02) and Poisson(1) probabilities are close. Zero-event error is 0.003710, actual total variation is 0.005583, and the independent upper bound is 0.020000.

Original exact calculations for a synthetic independent-indicator model. The upper chart shows counts 0–6; the lower chart uses one common absolute probability scale. The upper bound is a guarantee, not the realised error.

The Poisson-process constructions lesson develops the independent rare-event bound and the binomial-to-Poisson limit (membership; public preview).

What Chen–Stein adds

The method turns a probability error into a difference-equation calculation. For a count-event indicator $h(k)=\mathbf{1}_{\{k\in A\}}$ , solve the Poisson Stein equation

\lambda f(k+1)-k f(k)=h(k)-\mathbb{E}[h(Z)].

Substituting

W

and taking expectations makes the right-hand side exactly

\mathbb{P}(W\in A)-\mathbb{P}(Z\in A)

. Bounds on the solution, together with the structure of the indicators, control that error uniformly over

A

. This is the mechanism reviewed in Arratia, Goldstein and Gordon, §2.

For a finite collection, choose a neighbourhood $B_i$ containing $i$ . Define

\begin{aligned} b_1&=\sum_i\sum_{j\in B_i}p_i p_j,\\ b_2&=\sum_i\sum_{\substack{j\in B_i\j\ne i}}\mathbb{E}[X_iX_j],\\ b_3&=\sum_i\mathbb{E}\left[\left\lvert \mathbb{E}[X_i-p_i\mid\sigma(X_j:j\notin B_i)] \right\rvert\right]. \end{aligned}

A coarse form of their 1989 Theorem 1, converted to our convention, is

d_{\mathrm{TV}}(W,Z)\le b_1+b_2+b_3.

Here

b_1

measures neighbourhood probability mass,

b_2

joint occurrences inside it, and

b_3

residual dependence outside it. If

X_i

is independent of the whole collection outside

B_i

, then its contribution to

b_3

vanishes. Pairwise independence alone does not establish this. For mutually independent indicators, choose

B_i=\{i\}

b_2=b_3=0

, recovering the coarse independent bound.

A common shock can undo the approximation

Keep the same 50 marginal probabilities but set every

X_i=Y

, for one Bernoulli

(0.02)

variable

Y

. Then

W=50Y

still has mean one, but

\mathbb{P}(W=0)=0.98

. Its zero-event discrepancy from Poisson

(1)

is about

0.61212

, far above

0.02

. The independent bound's hypothesis has failed.

With $B_i$ equal to all 50 indices, $b_1=1$ , $b_2=49$ and $b_3=0$ : the dependent bound is valid but uninformative. In trading data, observed bursts therefore require an explicit dependence model and checked terms. Redefining a burst as one event changes what is counted; it does not automatically prove a Poisson approximation for either the burst count or the original trade count.

Sources

Lucien Le Cam, An Approximation Theorem for the Poisson Binomial Distribution, Pacific Journal of Mathematics 10 (1960), §3, Proposition 1; norm defined in §1.
R. Arratia, L. Goldstein and L. Gordon, Two Moments Suffice for Poisson Approximations: The Chen–Stein Method, Annals of Probability 17 (1989), §2, Theorem 1 and the discussion following Theorem 3.
R. Arratia, L. Goldstein and L. Gordon, Poisson Approximation and the Chen–Stein Method, Statistical Science 5 (1990), §2, equations (2)–(3).

What does the error bound mean?

Fifty opportunities, one expected event

What Chen–Stein adds

A common shock can undo the approximation

Sources

Related terms

Bernoulli distribution

Binomial distribution

Poisson distribution