Poisson theorem

2010 Mathematics Subject Classification: Primary: 60F05 [MSN][ZBL]

Poisson's theorem is a limit theorem in probability theory which is a particular case of the law of large numbers. Poisson's theorem generalizes the Bernoulli theorem to the case of independent trials in which the probability of appearance of a certain event depends on the trial number (the so-called Poisson scheme). Poisson's theorem states that: If in a sequence of independent trials an event $A$ occurs with probability $p _ {k}$ at the $k$- th trial, $k = 1 , 2 \dots$ and if $\mu _ {n} / n$ is the frequency of $A$ in the first $n$ trials, then for any $\epsilon > 0$ the probability of the inequality

$$\left | \frac{\mu _ {n} }{n} - \frac{p _ {1} + \dots + p _ {n} }{n} \right | < \epsilon$$

will tend to 1 when $n \rightarrow \infty$. Bernoulli's theorem follows from Poisson's theorem when $p _ {1} = \dots = p _ {n}$. The theorem was established by S. Poisson . The proof of Poisson's theorem was obtained by Poisson from a variant of the Laplace theorem. A simple proof of Poisson's theorem was given by P.L. Chebyshev (1846), who also stated the first general form of the law of large numbers, which includes Poisson's theorem as a particular case.

Poisson's theorem is a limit theorem in probability theory about the convergence of the binomial distribution to the Poisson distribution: If $P _ {n} ( m)$ is the probability that in $n$ Bernoulli trials a certain event $A$ occurs exactly $m$ times, where the probability of $A$ in every trial is $p$, then for large values $n$ and $1 / p$ the probability $P _ {n} ( m)$ is approximately

$$e ^ {- n p } \frac{( n p ) ^ {m} }{m!} .$$

The number $\lambda = n p$ is the mean number of occurrences of $A$ in $n$ trials, and the sequence of values $e ^ {- \lambda } \lambda ^ {m} / m !$, $m = 0 , 1 \dots$ $\lambda > 0$, forms a Poisson distribution. Poisson's theorem was established by S.D. Poisson [P] for a scheme of trials which is more general than the Bernoulli scheme, when the probability of occurrence of the event $A$ can vary from trial to trial so that $p _ {n} \rightarrow 0$ when $n \rightarrow \infty$. A strict proof of Poisson's theorem in this case is based on considering a triangular array of random variables so that in the $n$- th row the random variables are independent and take the values 1 and 0 with probability $p _ {n}$ and $1 - p _ {n}$, respectively. A more convenient form of Poisson's theorem is as an inequality: If $\lambda = p _ {1} + \dots + p _ {n}$, $\delta = p _ {1} ^ {2} + \dots + p _ {n} ^ {2}$, then when $n \geq 2$,

$$\left | P _ {n} ( m) - e ^ {- \lambda } \frac{\lambda ^ {m} }{m!} \right | \leq 2 \delta .$$

This inequality gives the error when $P _ {n} ( m)$ is replaced by $e ^ {- \lambda } \lambda ^ {m} / m !$. If $p _ {1} = \dots = p _ {n} = \lambda / n$, then $\delta = \lambda ^ {2} / n$. Poisson's theorem and Laplace's theorem give a complete description of the asymptotic behaviour of the binomial distribution. Subsequent generalizations of Poisson's theorem were made in two basic directions. On the one hand, further refinements of Poisson's theorem based on asymptotic expansions have emerged, and on the other hand general conditions have been established under which sums of independent random variables converge to a Poisson distribution.

References

 [P] S.D. Poisson, "Récherches sur la probabilité des jugements en matière criminelle et en matière civile" , Paris (1837) [L] M. Loève, "Probability theory" , Springer (1977) MR0651017 MR0651018 Zbl 0359.60001 [B] A.A. Borovkov, "Wahrscheinlichkeitstheorie" , Birkhäuser (1976) (Translated from Russian) MR0410818