Probability theory and stochastic processes Proof

Proof of the Poisson limit theorem

Builds on Binomial distribution Poisson distribution Convergence in distribution Cumulative distribution function Binomial coefficient 1 + x is at most exp x Logarithm is the inverse of the exponential

Proof

We split the binomial probability into three factors and find the limit of each one.

Fix an integer $k \ge 0$ . Since $n p_n \to \lambda$ , we have $p_n = (n p_n)/n \to \lambda \cdot 0 = 0$ . So there is $n_0 \ge k$ with $p_n \le 1/2$ for $n \ge n_0$ ; from now on $n \ge n_0$ .
By the definition of the binomial distribution and of the binomial coefficient, $P(S_n = k) = \binom{n}{k} p_n^k (1 - p_n)^{n-k} = a_n \cdot \frac{(n p_n)^k}{k!} \cdot (1 - p_n)^{n} \cdot (1 - p_n)^{-k},$ with $a_n = \frac{n(n-1)\cdots(n-k+1)}{n^k} = \prod_{j=0}^{k-1} \left(1 - \frac{j}{n}\right)$ .
Each of the $k$ factors $1 - j/n$ tends to $1$ . So $a_n \to 1$ , and for $k = 0$ the empty product is $1$ .
Since $n p_n \to \lambda$ , $(n p_n)^k / k! \to \lambda^k / k!$ . Since $p_n \to 0$ , $(1 - p_n)^{-k} \to 1$ .
Claim: $-x - x^2 \le \ln(1 - x) \le -x$ for $0 \le x \le 1/2$ . The upper bound is $1 - x \le e^{-x}$ , the inequality $1 + y \le e^y$ with $y = -x$ .
For the lower bound, let $g(x) = \ln(1 - x) + x + x^2$ . Then $g(0) = 0$ and $g'(x) = -\frac{1}{1-x} + 1 + 2x = \frac{x(1 - 2x)}{1 - x} \ge 0$ on $[0, 1/2]$ . So $g \ge 0$ there.
By step 5 with $x = p_n$ , $-n p_n - n p_n \cdot p_n \le n \ln(1 - p_n) \le -n p_n$ .
Both bounds tend to $-\lambda$ , since $n p_n \to \lambda$ and $n p_n \cdot p_n \to \lambda \cdot 0 = 0$ . So $n \ln(1 - p_n) \to -\lambda$ .
The exponential is continuous and is the inverse of the logarithm. So $(1 - p_n)^n = \exp(n \ln(1 - p_n)) \to e^{-\lambda}$ .
By steps 2, 3, 4 and 9, $P(S_n = k) \to 1 \cdot \frac{\lambda^k}{k!} \cdot e^{-\lambda} \cdot 1 = e^{-\lambda} \lambda^k / k!$ . This is the first claim.
Let $T \sim \operatorname{Poisson}(\lambda)$ , with the Poisson distribution. Let $F_n$ and $F$ be the cumulative distribution functions of $S_n$ and $T$ .
Both $S_n$ and $T$ take values in $\set{0, 1, 2, \dots}$ . So for $x < 0$ , $F_n(x) = 0 = F(x)$ .
For $x \ge 0$ , let $m = \floor{x}$ . Then $F_n(x) = \sum_{k=0}^{m} P(S_n = k)$ and $F(x) = \sum_{k=0}^{m} e^{-\lambda} \lambda^k / k!$ . These are finite sums, so step 10 gives $F_n(x) \to F(x)$ .
Thus $F_n(x) \to F(x)$ for every real $x$ , in particular at every continuity point of $F$ . So $S_n \to T$ in distribution.

Idea

For fixed $k$ , choosing $k$ successes among $n$ trials gives about $n^k/k!$ choices, each with probability about $p_n^k$ , so this part tends to $\lambda^k/k!$ . The chance of no other success, $(1 - p_n)^{n}$ , tends to $e^{-\lambda}$ because $\ln(1 - p) \approx -p$ for small $p$ .

Reading path

To understand this page from the ground up, read these 292 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, Proof that a function with nonnegative second derivative is convex, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof that a function with nonnegative derivative is increasing, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that the logarithm is the inverse of the exponential, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that the exponential function is positive, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof that a differentiable function is convex iff its derivative is increasing, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof that the exponential function is strictly increasing, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that 1 + x is at most exp x, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that slopes of a convex function are increasing, Proof that the exponential function is strictly convex, Proof of the constant multiple rule.

Binomial coefficient definition
Construction of the reals from Cauchy sequences concept
Convex function definition
Definition by recursion concept
Element definition
Factorial definition
Group concept
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Strictly monotone function definition
Proof of Pascal's rule from the recursive definition proof
Proof that slopes of a convex function are increasing proof
Strictly convex function definition
Set definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
Bernoulli trial concept
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Pascal's rule statement
Slopes of a convex function are increasing statement
Ordered pair definition
Commutative ring definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Union definition
Addition of natural numbers definition
Sample space definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Biconditional definition
The successor function is injective statement
Zero is not a successor statement
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Proof of the factorial formula for binomial coefficients by induction proof
Cartesian product definition
Continuous function definition
Differentiable function definition
One-sided derivative definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Subset definition
Order on the natural numbers definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Factorial formula for binomial coefficients statement
Proof that a composition of continuous functions is continuous proof
Binary relation definition
Derivative function definition
Order on the reals definition
Universal set definition
Power set definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Uniform convergence definition
Sum of a finite geometric series statement
Convergent series definition
Radius of convergence definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Higher derivative definition
Interval definition
Complement definition
Function definition
Strong induction statement
Uniform convergence on a set definition
Logical equivalence definition
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
Second derivative definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Sigma-algebra definition
Injective function definition
Preimage of a set definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Monotone function definition
Epsilon-delta definition of a limit statement
Countable set definition
Event definition
Well-ordering principle statement
Borel sigma-algebra definition
Double negation law statement
Total order definition
Composition of continuous functions is continuous statement
Probability axioms concept
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Probability measure definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Probability distribution definition
Probability space definition
The reals form an ordered field statement
Absolute value definition
Random variable definition
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Cumulative distribution function definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Discrete random variable definition
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Existence of the infimum statement
Convergence in distribution definition
Triangle inequality statement
Absolute value is multiplicative statement
Probability mass function definition
Proof of the intermediate value theorem by the supremum proof
Proof that one-sided limits of a monotone function exist proof
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof of the uniqueness of limits proof
Proof of the limit of a sum of sequences proof
Poisson distribution definition
Binomial distribution definition
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof of the term test proof
Proof that Cauchy sequences are bounded proof
The sequence 1/n tends to zero example
Proof of the monotone convergence theorem for sequences proof
Differentiable functions are continuous statement
Proof of the continuity of the inverse of a strictly monotone continuous function proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limits preserve non-strict inequalities statement
Uniqueness of limits statement
Limit of a sum of sequences statement
Proof of the ratio formula for the radius of convergence proof
Convergent sequences are bounded statement
Squeeze theorem statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
One-sided limits of a monotone function exist statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Continuity of the inverse of a strictly monotone continuous function statement
Proof of the sequential characterization of limits proof
Decreasing sequence bounded below converges statement
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Geometric series example
Continuity is equivalent to sequential continuity statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the ratio test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Proof of the power rule by induction proof
Boundedness theorem statement
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a differentiable function is convex iff its derivative is increasing proof
Proof that a function with nonnegative derivative is increasing proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Proof that a function with positive derivative is strictly increasing proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
A function with nonnegative derivative is increasing statement
A differentiable function is convex iff its derivative is increasing statement
Uniform convergence of derivatives allows term-by-term differentiation statement
A function with positive derivative is strictly increasing statement
Proof that the exponential series has infinite radius of convergence proof
Proof that a function with nonnegative second derivative is convex proof
Proof of term-by-term differentiation proof
The exponential series has infinite radius of convergence statement
A function with nonnegative second derivative is convex statement
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
The exponential function is positive statement
The exponential function is its own derivative statement
Proof that the exponential function is strictly increasing proof
The exponential function is strictly increasing statement
Proof that the exponential function is strictly convex proof
Natural logarithm definition
Proof that 1 + x is at most exp x proof
The exponential function is convex statement
1 + x is at most exp x statement
Proof that the logarithm is the inverse of the exponential proof
Logarithm is the inverse of the exponential statement
Proof of the Poisson limit theorem proof

Where this leads

1 page builds on this one.

Poisson limit theorem

Tag	`PCFHW6Y6ZQ`
URI	`m2://PCFHW6Y6ZQ`
Kind	proof
Writing	W0 stub
MSC	60F05
Proves	Poisson limit theorem
Uses	Binomial distribution, Poisson distribution, Convergence in distribution, Cumulative distribution function, Binomial coefficient, 1 + x is at most exp x, Logarithm is the inverse of the exponential
Changed	4 November 2026

Proof of the Poisson limit theorem

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here