<P> If Σ (\ displaystyle \ Sigma) is a p × p (\ displaystyle p \ times p) positive - semidefinite covariance matrix with strictly positive diagonal entries, then for independent and identically distributed X, Y ∼ N (0, Σ) (\ displaystyle X, Y \ sim N (0, \ Sigma)) and any random p (\ displaystyle p) - vector w (\ displaystyle w) independent of X (\ displaystyle X) and Y (\ displaystyle Y) such that w 1 + ⋯ + w p = 1 (\ displaystyle w_ (1) + \ cdots + w_ (p) = 1) and w i ≥ 0, i = 1, ⋯, p, (\ displaystyle w_ (i) \ geq 0, i = 1, \ cdots, p,) it holds that </P> <Dl> <Dd> ∑ j = 1 p w j X j Y j ∼ C a u c h y (0, 1). (\ displaystyle \ sum _ (j = 1) ^ (p) w_ (j) (\ frac (X_ (j)) (Y_ (j))) \ sim \ mathrm (Cauchy) (0, 1).) </Dd> </Dl> <Dd> ∑ j = 1 p w j X j Y j ∼ C a u c h y (0, 1). (\ displaystyle \ sum _ (j = 1) ^ (p) w_ (j) (\ frac (X_ (j)) (Y_ (j))) \ sim \ mathrm (Cauchy) (0, 1).) </Dd> <P> If X 1,..., X n (\ displaystyle X_ (1),..., X_ (n)) are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean (X 1 +...+ X n) / n (\ displaystyle (X_ (1) +...+ X_ (n)) / n) has the same standard Cauchy distribution . To see that this is true, compute the characteristic function of the sample mean: </P>

Find the cdf of cauchy random variable which has the pdf