probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, Le Cam's theorem, named after

Lucien Le Cam Lucien Marie Le Cam (November 18, 1924 – April 25, 2000) was a mathematician and statistician. Biography Le Cam was born November 18, 1924 in Croze, France. His parents were farmers, and unable to afford higher education for him; his father died ...

(1924 – 2000), states the following. Suppose: *

X_1, X_2, X_3, \ldots

are

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

s, each with a

Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probab ...

(i.e., equal to either 0 or 1), not necessarily identically distributed. *

\Pr(X_i = 1) = p_i, \text i = 1, 2, 3, \ldots.

\lambda_n = p_1 + \cdots + p_n.

S_n = X_1 + \cdots + X_n.

(i.e.

S_n

follows a Poisson binomial distribution) Then :

\sum_^\infty \left,  \Pr(S_n=k) -  \ < 2 \left( \sum_^n p_i^2 \right).

In other words, the sum has approximately a

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known ...

and the above inequality bounds the approximation error in terms of the

total variation distance In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance, statistical difference or variational dist ...

. By setting ''p''_''i'' = λ_''n''/''n'', we see that this generalizes the usual Poisson limit theorem. When

\lambda_n

is large a better bound is possible:

\sum_^\infty \left,  \Pr(S_n=k) -  \ < 2 \left(1 \wedge \frac 1 \lambda_n\right) \left( \sum_^n p_i^2 \right).

It is also possible to weaken the independence requirement.

References

External links

* {{MathWorld, urlname=LeCamsInequality, title=Le Cam's Inequality Probability theorems Probabilistic inequalities Statistical inequalities Theorems in statistics