MathDB

Let

Y_n

be a binomial random variable with parameters

n

and

p

. Assume that a certain set

H

of positive integers has a density and that this density is equal to

d

. Prove the following statements: (a) \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d if

H

is an arithmetic progression. (b) The previous limit relation is not valid for arbitrary

H

. (c) If

H

is such that

P(Y_n \in H)

is convergent, then the limit must be equal to

d

. L. Posa

Problem Statement