3. Renewal Limit Theorems
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1 Virul Lborories > 14. Renewl Processes > Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process is he pril sum process for sequence of independen, ideniclly disribued vribles (he inerrrivl imes). Thus, i seems resonble h he fundmenl limi heorems for pril sum processes (he lw of lrge numbers nd he cenrl limi heorem heorem), should hve nlogs for he couning process. Th is indeed he cse, nd he purpose of his secion is o explore he limiing behvior of renewl processes. The min resuls h we will sudy, known ppropriely enough s renewl heorems, re imporn for oher sochsic processes, priculrly Mrkov chins. Thus, consider renewl process wih inerrrivl disribuion F nd men μ, wih he ssumpions nd bsic noion esblished in he inroducory secion. When μ =, we le 1 μ = 0. When μ <, we le σ denoe he sndrd deviion of he inerrrivl disribuion. Bsic Theory A Lw of Lrge Numbers 1. Suppose h μ <. Show h N rrivls per uni im 1 μ s wih probbiliy 1. Thus, 1 μ is he limiing verge re of Recll h T N < T N +1 for > 0 Hence T N T N +1 < when N N N N > 0 Recll h N s wih probbiliy 1. Conclude h N +1 N 1 s wih probbiliy 1. Recll h by he srong lw of lrge numbers, T n n μ s n wih probbiliy 1. A Cenrl Limi Theorem The purpose of his prgrph is o show h he couning vrible N is sympoiclly norml. Thus, suppose h μ nd σ re finie, nd le Z = N μ /, > 0 σ / μ 3 2. Show h he disribuion of Z converges o he sndrd norml disribuion s. Le W n = T n n μ for n N + nd recll h he disribuion of W n converges o he sndrd norml disribuion σ n
2 g. s n, by he ordinry cenrl limi heorem. Show h for z R, P( Z z) = P(T n( z, ) > ) where n( z, ) = μ + z σ μ 3. z Show h P( Z z) = P(W n( z, ) > w( z, )) where w( z, ) =. Show h n( z, ) s Show h w( z, ) z s 1+( z σ ) / μ Recll h 1 Φ( z) = Φ( z), where s usul, Φ is he sndrd norml disribuion funcion. Conclude h P( Z z) Φ() s The Elemenry Renewl Theorem The Elemenry Renewl Theorem ses h he bsic limi in he lw of lrge numbers bove holds in men, s well s wih probbiliy 1. Th is, he limiing men verge re of rrivls is 1 μ : m() 1 μ s The elemenry renewl heorem is of fundmenl impornce in he sudy of he limiing behvior of Mrkov chins. The proof, skeched in he following exercises, is no nerly s esy s one migh hope (recll h convergence wih probbiliy 1 does no imply convergence in men). 3. Show h lim inf m() 1 μ. Noe firs h he resul is rivil if μ =, so ssume h μ <. Show h N + 1 is sopping ime for he sequence of inerrrivl imes X. Recll h T N +1 > for > 0. Use Wld's equion o show h (m() + 1) μ > Conclude h m() > 1 μ 1 for > 0. For he nex pr of he proof, we will runce he rrivl imes, nd use he bsic comprison meho For > 0, le X, i = { X i, X i, X i > nd consider he renewl process wih he sequence of inerrrivl imes X = ( X, 1, X, 2,...). We will use he sndrd noion developed in he inroducory secion.. 4. Show h lim sup m() 1 μ. Show h T, N, +1 + for > 0 nd for > 0. Use Wld's equion gin o show h (m () + 1) μ +
3 g. Conclude h m () 1 ( μ + μ ) 1 Recll h m() m () for > 0 nd > 0. Conclude h m() ( 1 μ + μ ) 1 for > 0 nd > 0. for > 0 nd > 0. m() Conclude h lim sup 1 μ. for > 0. Use he monoone convergence heorem o show h μ μ s. The Renewl nd Key Renewl Theorems This secion gives he deepes nd mos useful of he limi heorems in renewl heory. The proofs re rher compliced nd re omie Suppose h he renewl process is periodi The renewl heorem ses h, sympoiclly, he expeced number of renewls in n inervl is proporionl o he lengh of he inervl; he proporionliy consn is 1 μ. Specificlly, for every h 0, m( (, + h] ) h μ s The renewl heorem is lso known s Blckwell's heorem in honor of Dvid Blckwell. The key renewl heorem is n inegrl version of he renewl heorem. Suppose gin h he renewl process is periodic nd suppose h g is decresing funcion from [ 0, ) o [ 0, ) wih 0 g()d <. Then 1 0 g( x)dm( x) μ 0 g( x)d x s The key renewl heorem cn be exended o more generl clss of funcions known s direcly Riemnn inegrble funcions. The nme, of course, refers o Georg Riemnn. See Sochsic Processes by Sheldon Ross for more deils. 5. Use he renewl heorem o prove he elemenry renewl heorem: Le n = m( ( n, n + 1] ). for n N. Use he renewl heorem o show h n 1 μ s n. Conclude h 1 n n 1 k k 1 =0 μ s n. Conclude h m(n) n 1 μ s n. Use he fc h he renewl funcion m is incresing o show h for > 0, m( ) m() m( ) Use he squeeze heorem for limis o conclude h m() 1 s. μ 6. Conversely, he elemenry renewl heorem lmos implies he renewl heorem. Assume h g( x) = lim m( + x) m() exiss for ech x 0. Noe h m( + x + y) m() = (m( + x + y) m( + x)) + (m( + x) m()) Le o conclude h g( x + y) = g( x) + g( y) for ll x 0 nd y 0.
4 Show h g is incresing. Conclude h g( x) = c x for x 0 where c is consn. Excly s in prs ()-(c) of he previous exercise, show h m(n) n c s n. From he elemenry renewl heorem, conclude h c = 1 μ. 7. Show h he key renewl heorem implies he renewl heorem: pply he key renewl heorem o g h ( x) = 1(0 x h) where h 0. Conversely, he renewl heorem implies he key renewl heorem. Exmples nd Specil Cses The Poisson Process Recll h he Poisson process, he mos imporn of ll renewl processes, hs inerrrivl imes h re exponenilly disribued wih re prmeer r > 0. Thus, he inerrrivl disribuion funcion is F( x) = 1 e r x for x 0 nd he men inerrrivl ime is μ = 1 r. 8. Verify ech of he following direcly: The lw of lrge numbers for he couning process. The cenrl limi heorem for he couning process. The elemenry renewl heorem. The renewl heorem. Bernoulli Trils Suppose h X = ( X 1, X 2,...) is sequence of Bernoulli rils wih success prmeer p ( 0, 1). Recll h X is sequence of independen, ideniclly disribued indicor vribles wih p = P( X = 1). We hve sudied number of rndom processes derived from X: Y = (Y 0, Y 1,...) where Y n he number of success in he firs n rils. The sequence Y is he pril sum process ssocied wih X. The vrible Y n hs he binomil disribuion wih prmeers n nd p. U = (U 1, U 2,...) where U n he number of rils needed o go from success number n 1 o success number n. These re independen vribles, ech hving he geomeric disribuion wih prmeer p. V = (V 0, V 1,...) where V n is he ril number of success n. The sequence V is he pril sum process ssocied wih U. The vrible V n hs he negive binomil disribuion wih prmeers n nd p.
5 9. Consider he renewl process wih inerrrivl sequence U. Thus, μ = 1 p is he men inerrrivl ime, nd Y is he couning process. Verify ech of he following direcly: The lw of lrge numbers for he couning process. The cenrl limi heorem for he couning process. The elemenry renewl heorem. 10. Consider he renewl process wih inerrrivl sequence X. Thus, he men inerrrivl ime is μ = p. nd he number of rrivls in he inervl [ 0, n] is V n +1 1 for n N. Verify ech of he following direcly: The lw of lrge numbers for he couning process. The cenrl limi heorem for he couning process. The elemenry renewl heorem. Virul Lborories > 14. Renewl Processes > Conens Apples D Ses Biogrphies Exernl Resources Keywords Feedbck
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