First note that for all K > 0,

where the last equation follows from the inequality

with r = ∞ and Y_t = X_t I(|X_t| ≤ K) and the inequality

Next, note that for all K ≥ 1,

for some constant C_p1, where the first inequality is the Markov inequality, implying that for all K ≥ 1,

Now setting K = 2pE exp(|X_t|)log(1 + α(m)⁻¹) (and noting that K > 1) will turn the preceding inequality into

Now note that E exp(|X_t|) > 1, implying that the last expression can be bounded by

for constants C_p2 and C_p3. Setting C_p = 12p + C_p3 now gives

which is the stated result.