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Lemma 9.11.13A56815B99E6B

Lemma 9.11.13.A56815B99E6B Let $A,B,\alpha ,\beta ,T>0$. Then

\[ \int _ T^\infty \int _ T^\infty \exp \bigl (-\alpha p-\beta q+\min (Ap,Bq)\bigr ) \, \mathrm{d}q\, \mathrm{d}p<\infty \quad \Longleftrightarrow \quad \frac\alpha A+\frac\beta B>1. \]

At equality the integral diverges.

Proof. Split the quadrant along $q=Ap/B$. On the part $q\geq Ap/B$, set $p_0=\max \{ T,BT/A\} $. After discarding the strip $T<p<p_0$, integration in $q$ gives

\[ \frac1\beta \int _{p_0}^\infty \exp \left(-\left[\alpha -A+\frac{\beta A}{B}\right]p\right) \, \mathrm{d}p. \]

The discarded strip has finite integral because $\beta >0$. On the other part, integration in $p$ gives the corresponding tail with coefficient $\beta -B+\alpha B/A$; its discarded strip is finite because $\alpha >0$. Both coefficients are positive exactly when $\alpha /A+\beta /B>1$. If that sum equals one, either tail is an integral of a positive constant over a half-line. $\square$

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