On the large-sample behavior of two estimators of the conditional copula under serially dependent data

The conditional copula of a random pair $$(Y_1,Y_2)$$ ( Y 1 , Y 2 ) given the value taken by some covariate $$X \in {\mathbb {R}}$$ X ∈ R is the function $$C_x:[0,1]^2 \rightarrow [0,1]$$ C x : [ 0 , 1 ] 2 → [ 0 , 1 ] such that $${\mathbb {P}}(Y_1 \le y_1, Y_2 \le y_2 | X=x) = C_x \{ {\mathbb {P}}(Y_1\le y_1 | X=x), {\mathbb {P}}(Y_2\le y_2 | X=x) \}$$ P ( Y 1 ≤ y 1 , Y 2 ≤ y 2 | X = x ) = C x { P ( Y 1 ≤ y 1 | X = x ) , P ( Y 2 ≤ y 2 | X = x ) } . In this note, the weak convergence of the two estimators of $$C_x$$ C x proposed by Gijbels et al. (Comput Stat Data Anal 55(5):1919–1932, 2011) is established under $$\alpha $$ α -mixing. It is shown that under appropriate conditions on the weight functions and on the mixing coefficients, the limiting processes are the same as those obtained by Veraverbeke et al. (Scand J Stat 38(4):766–780, 2011) under the i.i.d. setting. The performance of these estimators in small sample sizes is investigated with simulations.