Definition 1. An m × 1 random vector X is said to have an elliptical distribution with parameter µ (m × 1) and V (m × m) if its density function has the form:

3

2. Preliminary Results

First we introduce the definition of elliptical distribution, which we will use to develop our methods in the sequent.

Definition 1. An m × 1 random vector X is said to have an elliptical distribution with parameter µ (m × 1) and V (m × m) if its density function has the form:

c m (det V ) ^1/2 h ¡

(x − µ) ^´ V ⁻¹ (x − µ) ¢

for some function h, where c m is a normalizing constant, and V is positive definite.

To achieve our goal, first we need to study the asymptotic distribution of r. The following Theorems 2 and 3 are the well-known results for this, which can be found, for example, in Muirhead (1982), section 5.1.

Theorem 2. Let r be the correlation coefficient formed from a sample of size n from a bivariate elliptical distribution with correlation coefficient ρ and kurtosis parameter κ. Then the asymptotic distribution, as n → ∞, of √

n − 1 _1−ρ ^r−ρ

is N(0, 1 + κ) .

Theorem 2 states that r is asymptotically normal as n → ∞. A more accurate result can be obtained by considering z = tanh ⁻¹ r = ¹ ₂ log ^1+r _1−r . Fisher (1921) showed that the distribution of z converges to the normal distribution much faster than r itself.

Theorem 3. Let r be the correlation coefficient formed from a sample of size n, n > 1, from a bivariate elliptical distribution with correlation coefficient ρ and kurtosis parameter κ. Define z=tanh ⁻¹ r= ¹ ₂ log ^1+r _1−r and ξ=tanh ⁻¹ ρ= ¹ ₂ log ^1+ρ _1−ρ . Then, as n → ∞, the asymptotic distribution of √

n − 1 (z − ξ) is N (0, 1 + κ) .

Proof. We use the fact that if {X ⁿ } is a sequence of random variables such that √

n − 1 (X ⁿ − µ) → N (0, σ ² ) in distribution as n → ∞, and if f (x) is a function which is differentiable at x = µ, then √

n − 1 {f (X ⁿ ) − f (µ)} → N ¡

0, f ^´ (µ) ² σ ² ¢

in distribution as n → ∞ ; see, e.g., Bichel

and Doksum (1977), p.461. Let f (x) = ¹ ₂ log ^1+x _1−x . Note that f ^´ (x) ² (1 − x ² ) ² = 1. According

to Theorem 2,

4

√ n − 1 (r − ρ) → N ³

0, (1 + κ) ¡

1 − ρ ² ¢ 2 ´

as n → ∞, and hence

√ n − 1 {f (r) − f (ρ)} → N ³

0, f ^´ (ρ) ² (1 + κ) ¡

1 − ρ ² ¢ 2 ´

as n → ∞,

namely

√ n − 1 (z − ξ) → N (0, (1 + κ)) as n → ∞

¤