Approximations in Bivariate Renewal Theory

Edward Omey, Kosto Mitov, Rein Vesilo

We construct approximations to the renewal function for a bivariate renewal process. Suppose $(X,Y)$, $(X_1,Y_1)$, $(X_2,Y_2),\ldots$ denote i.i.d. positive random vectors with common distribution function $F(x,y)=\operatorname{P}(X\leq x,Y\leq y)$. Let $S_n^{(1)}=X_1+X_2+\cdots+X_n$ and $S_n^{(2)}=Y_1+Y_2+\cdots+Y_n$ denote the partial sums where we set $S_0^1=S_0^2=0$. Associated with $\{(X_i,Y_i)\}$, we define, respectively, the univariate and bivariate renewal counting processes: $N_i(x)=\min\{n\geq 1:S_n^{(i)}>x\}$ ($i=1,2$) and $N(x,y)=\min\{N_1(x),N_2(y)\}$. The bivariate renewal function is given by $U(x,y)=\operatorname{E} N(x,y)=\sum_0^{\infty}F^{*n}(x,y)$. From the practical point of view it is hard to find explicit expressions for the renewal function $U(x,y)$. Recently, Mitov and Omey (2014) introduced a new method to obtain approximations for univariate renewal functions based on expansions of Laplace--Stieltjes transforms. In this paper we generalize these approximations to the bivariate case and apply them to regularly varying increments. We show that the approximation along the diagonal is different from off the diagonal.