content_image/317271.jpg content_image/317272.jpg content_image/317273.jpg
Under the null, 

<img src='content_image/317271.jpg'>

where the first inequality is the Bonferroni inequality, and the second inequality fol- lows from equation (A.8). Since the last term goes to 0 as N → ∞ if lim$_{N}$$_{→∞}$ m/N 1 / 3 = 0, the proof of item 1 is complete. 

The proof of item 2 is very similar to the proof in Section A.1, the only modification is an additional application of Bonferroni’s inequality under the null: 

<img src='content_image/317272.jpg'>

where the first inequality in the last row follows from (A.12). Since m$_{max}$ ( N m$_{max}$ ) is at most O ( N × N$^{√}$ $^{N}$), and since 

it follows that the last expression goes to zero as N → ∞ . 

<img src='content_image/317273.jpg'>

## D An example of mutual information estimation 

We examined our suggested estimator in the following setup. For m = 15 and N ∈ { 300 , 1000 } sample points drawn from a two–component Gaussian mixture, we 

