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inf$_{x}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ d | f$_{X}$ ( x ) | > 0 and Fubini’s theorem 

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A change of variables argument yields ∫ [0 , 1] | ϕ$_{J,k}$$_{l}$ ( x$_{l}$ ) | d µ ( x$_{l}$ ) ≳ 2 − J/ 2 uniformly for 0 ≤ k$_{l}$ ≤ 2 J − 1, and so E [ | b$_{Kk}$ ( X ) | ] ≳ 2 − dJ/ 2 = K − 1 / 2 uniformly for 1 ≤ k ≤ K . 

For the i.i.d. case , define b ∗ $_{Kk}$( X$_{i}$ ) = n − $^{1}$( | b$_{Kk}$ ( X$_{i}$ ) | − E [ | b$_{Kk}$ ( X$_{i}$ ) | ]) / ( E [ | b$_{Kk}$ ( X$_{i}$ ) | ]) for each 1 ≤ k ≤ K . It may be deduced from the preceding bounds and the fact that E [ b$_{Kk}$ ( X$_{i}$ ) $^{2}$] ≍ 1 that ‖ b ∗ Kk ‖ ∞ ≲ K/n and E [ b ∗ $_{Kk}$( X$_{i}$ ) $^{2}$] ≲ K/n $^{2}$. By the union bound and Bernstein’s inequality (see, e.g., pp. 192–193 of Pollard (1984)) we obtain, for any t > 0, 

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where c$_{1}$ and c$_{2}$ are finite positive constants independent of t . The right-hand side of (141) vanishes as n → ∞ since K log n/n = o (1). 

For the beta-mixing regressors case , we may extend the proof for the i.i.d. case using a coupling argument similar to the proof of Theorem 4.2 to deduce that 

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The right-hand side is o (1) provided n $_{q}$β ( q ) = o (1) and ( qK log n ) /n = o (1). Both these conditions are satisfied under the conditions on K , taking q = γ − $^{1}$log n in the exponentially β -mixing case and q ≍ n γ/ (1+ γ ) in the algebraically β -mixing case. 

