ABSTRACT
The chapter is restricted to counting models P with support in N0 = {0,1,2, . . .}. The density p(k) with respect to the counting measure is simply p(k) = P({k}),k ∈ N0. Given a sample xn ∈ Nn0 the empirical density is given by
pˆn(k) = 1 n
∑ i=1 {xi = k} , k ∈ N0 . (3.1)
For data xn = Xn = Xn(P) generated under the model it is seen that npˆn(k) ∼ b(n, p(k)). This implies
E(pˆ(k)) = p(k), V(pˆ(k)) = p(k)(1− p(k))
n (3.2)
and
E((pˆ(k)− p(k))(pˆ(`)− p(`))) =− p(k)p(`) n
, 0≤ k < `, . . . (3.3)
The by √
n
where (W
E(W (k)W (`)) = −
p(k)p(`) (1− p(k))(1− p(`)) , 0≤ k < ` < ∞ . (3.6)
The (W (k))∞k=0 process can be constructed as follows. Let (Z(k)) ∞ k=0 be a standard
white-noise Gaussian process and put
W = ∞
√ p(k) Z(k), (3.7)
W (k) = (Z(k)− √
p(k)W )/ √
1− p(k) . (3.8)
The sum in (3.7) converges in mean square so that the random variable W is well defined.
