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By Julian Lowell Coolidge

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S/ is the characteristic function of the interval. s/ is directly Riemann integrable on Œ0; 1/ for all " small enough, say " Ä "2 . s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/. 1/. "/ Condition M2 implies that m1 < 1 for all " small enough, say " Ä "3 . 36) The renewal theorem implies the following statement. 3. Let conditions D3 , M2 , and F3 hold. t / ! 1/ as t ! s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/.

Condition D2 implies condition D5 . "/ ! 1 as " ! 0. s/. 9 Convergence of moments for perturbed renewal processes In this section we give conditions for convergence of moments for perturbed renewal processes. 5. Let conditions D2 and M2 hold. "/ ! 1 as " ! 0/ ! m1 / r as " ! "/ Proof. Condition D2 implies condition D5 . 47), the theorem follows from an appropriate variant of the Lebesgue theorem. 46) can be rewritten in an equivalent form, which is actually a triangular array analogue of the elementary renewal theorem.

1/. "/ Condition M2 implies that m1 < 1 for all " small enough, say " Ä "3 . 36) The renewal theorem implies the following statement. 3. Let conditions D3 , M2 , and F3 hold. t / ! 1/ as t ! s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/. This is true, since the formula involves the Lebesgue integration. "/ . 1/ as t ! 1. 37). 4. Let conditions M2 , and F2 hold. 1/ ! 1/ as " ! 0/ Proof. "/ 1 ! 0; 1/ as " ! 0. ds/ ! ds/ < 1 as " ! ds/ ! ds/ as " ! 4.

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An introduction to mathematical probability by Julian Lowell Coolidge


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