Stat 5101 Lecture Notes - School of Statistics

72 Stat 5101 (Geyer) Course Notesunbounded domain of integration. The second integral on the right hand sidemust be finite: the integral of a bounded function over a bounded domain isalways finite, we do not need to check.It is rare that we can exactly evaluate the integrals. Usually we have to useTheorem 2.36 to settle the existence question by comparing with a simpler integral.The following lemmas give the most useful integrals for such comparisons.While we are at it, we give the analogous useful infinite sums. The proofs areall elementary calculus.Lemma 2.39. For any positive real number a or any positive integer m∫ ∞exist if and only if b−1.∫ a0x b dxLemma 2.41. For any positive real number a or any positive integer m andany positive real number c and any real number b (positive or negative)∫ ∞ax b e −cx dxand∞∑n b e −cnn=mexist.The following two lemmas give us more help using the domination theorem.Lemma 2.42. Suppose g and h are bounded, strictly positive functions on aninterval [a, ∞) andg(x)lim = k, (2.52)x→∞ h(x)where k is a strictly positive constant, then either both of the integrals∫ ∞ag(x) dxand∫ ∞are finite, or neither is. Similarly, either both of the sumsah(x) dx (2.53)∞∑g(k)k=mand∞∑h(k) (2.54)k=mare finite, or neither is, where m is any integer greater than a.