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It establishes that the key CLT and Proposition 6 remain valid under perturbations of random arm returns, including the case τ = 0, and further generalizes the payoff test function u by proving the result when u is continuous with controlled polynomial growth. The appendix provides stepwise proofs, including reductions from compactly supported u to general growth-controlled u, and uses PDE/HJB characterizations, comparison estimates, and moment bounds to control approximation error.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/approximate-optimality-and-the-riskreward-tradeo-given-repeated-gambles-online-appendix/268772/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/approximate-optimality-and-the-riskreward-tradeo-given-repeated-gambles-online-appendix/268772.png","ImageObject",300,407,{"name":92,"@type":93},"\tCallum ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the appendix prove about Proposition 6 and the CLT when τ = 0?","Question",{"text":112,"@type":113},"It shows the CLT and Proposition 6 remain valid when τ = 0 by adding a small perturbation to the random returns of the K arms and controlling the resulting change in the objective via bounds derived from the associated PDE solution.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is V_{τ,n} defined in this appendix?",{"text":117,"@type":113},"V_{τ,n} is defined through a supremum over squared terms involving expectations of u evaluated on the perturbed reward pair (Z^1_{τ,n}, Z^2_{τ,n}). It is then related to logarithmic asymptotics and the solution of a PDE with a modified function G_{τ}.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the appendix extend validity from compactly supported u to general continuous u?",{"text":121,"@type":113},"It proves the CLT first for u with compact support, then decomposes a general growth-controlled u into u1 (compact support) and u2 (vanishing outside a ball of radius N). It bounds the contribution of u2 using moment inequalities and the Burkholder–Davis–Gundy inequality to ensure the error can be made arbitrarily small.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},268772,1789436594,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143,"preview_page_count":4},137451211410,"https://ap-avatar.wpscdn.com/avatar/2000bb0a9246f588df?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786362646172706240","Journal: Economic Theory  \n\"Approximate optimality and the risk/reward tradeo§ given repeated gambles\"Authors: Zengjing Chen, Larry G Epstein, and Guodong Zhang  \nCorresponding author: Larry G Epstein, McGill University, [larry.epstein@mcgill.ca](larry.epstein@mcgill.ca)  \nONLINE APPENDIX  \nLemma: Our CLT, Proposition 6, is valid also if 􀀛 = 0 .  \nProof: As in the proof of Proposition 6, it su¢ ces to take u 2C(R2 ) . Given 􀀛 = 0, we add a perturbation to the random returns of the K arms.  \nFor any 1 􀀔 k 􀀔 K and n 􀀕 1, let X􀀏k;n = Xk;n + 􀀏􀀐n , where 􀀏 > 0 is a Öxed small constant and f􀀐ng is a sequence of i.i.d. standard normal random variables, independent with fXk;ng. Then, for any 􀀒 2 􀀂 and n 􀀕 1, the corresponding reward is denoted by Z􀀒;􀀏n = Z􀀒n + 􀀏􀀐n , and the corresponding set of mean-variance pairs is denoted by  \nA􀀏 = f (􀀖k;􀀏 ; 􀀛 2k;􀀏) : 1 􀀔 k 􀀔 Kg ;  \nwhere 􀀖 k;􀀏 = 􀀖 k and 􀀛 2k;􀀏 = 􀀛 2k + 􀀏 2 . The corresponding bounds are 􀀖􀀏 ; 􀀖􀀏 ; 􀀛  2􀀏 , and 􀀛  2􀀏 > 0.  \nDeÖne  \nV􀀏n = sup EP 􀀒2􀀂  \n\"u  \nP~~1~~n~~ ~~Z~~i~~􀀒;􀀏 ; P~~ ~~(Zi􀀒;􀀏~~ ~~􀀀Ep~~ n~~P~~ ~~[Zi􀀒;􀀏~~ ~~jH􀀒i􀀀1] !\\#  \nBy Proposition 6 for fZ􀀒;􀀏ng,  \nln V􀀏n = a2[suA􀀏]p(0 ; 1) EP 􀀔u 􀀒Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) 􀀓􀀕 = v􀀏 (0 ; 0 ; 0) ; (43)  \nwhere v􀀏 (t; x; y) is the solution of PDE (20) with function G􀀏 instead of G,  \nG􀀏 (p; q) = (􀀖;s􀀛u2)p2A􀀏 􀀔􀀖p + 12􀀛 2 q􀀕 , (p; q) 2 R2 . (44)  \nBy Yong and Zhou (1999, Propn. 5.10, Ch. 4), 9C0 > 0 such that jv􀀏 (t; x; y) 􀀀 v(t; x; y)j 􀀔 C0 p􀀏 ; 8 (t; x; y) 2 [0 ; 1) 􀀂 R2 :  \nWe also have  \njVn 􀀀 V􀀏n j2 􀀔 C􀀏2 EP \" 􀀌 P~~ ~~􀀐i 􀀌 2 + 􀀌 P~~1n ~~􀀐i 􀀌 2\\# 􀀔 2C􀀏2 ,  \nwhere the constant C depends only on the bounds of @xu and @yu.  \nLetting as 􀀏 ! 0 in (43), the CLT (25) is proven for 􀀛 = 0 . Similar arguments  \nshow that (26) is also valid. 􀀄  \nLemma: Our CLT, Proposition 6, is valid also if u is continuous and, for some g 􀀕 1 and c > 0 , ju(x; y)j 􀀔 c(1 + jj (x:y)jjg􀀀1) and sup1􀀔k􀀔K EP [jXk jg ] \u003C 1 .  \nProof: We prove that (25) remains valid. Refer to it as \"the CLT. \"  \nStep 1: Prove the CLT for any u 2 Cb (R2 ) with compact support (constant outside a compact subset of R2 ) . In this case, 8􀀏 > 0 9u^ 2 Cb;Lip (R2 ) such that supz2R2 ju(z) 􀀀 u^(z)j 􀀔 ~~􀀏~~2 . Then  \n􀀌 s􀀒p􀀂 EP \"u ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2 s[uA]p(0 ; 1) EP [u(Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \n􀀔􀀏 + 􀀌 s􀀒p􀀂 EP \"u^ ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2s[uA]p(0 ; 1) EP [u^(Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \nTherefore,  \n􀀌  \n􀀌 lim sup 􀀌 n!1 􀀌  \nsup EP  \n􀀒2􀀂  \n\"u ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2s[uA]p(0 ; 1) EP [u(Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌 􀀔 􀀏;  \nwhich proves the CLT since 􀀏 is arbitrary.  \nStep 2: Let u 2 C (R2 ) satisfy the growth condition ju(z)j 􀀔 c(1 + jz jg􀀀1) for g 􀀕 1. For any N > 0 , 9u1 ; u2 2C(R2 ) such that u = u 1 + u2 , where u 1 has a compact support and u2 (z) = 0 for jz j 􀀔 N , and ju2 (z)j 􀀔 ju(z)j for all z. Then  \n2c(1 + jz jg ) ju2 (z)j 􀀔 N ; 8z 2 R2 ,  \nand  \n􀀌  \n􀀌  \n􀀌  \n􀀌  \n􀀌  \n􀀌  \n􀀌  \n􀀔 􀀌  \n􀀌  \n􀀌  \nsup EP  \n􀀒2􀀂  \nsup EP  \n􀀒2􀀂  \n\"u  \n\"u 1  \nSn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2 s[uA]p(0 ; 1) EP [u(Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \nSn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2s[uA]p(0 ; 1) EP [u1 (Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \n+ sup EP  \n􀀒2􀀂  \n\" 􀀌 u2 ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ ! 􀀌 \\# + a2s[uA]p(0 ; 1) EP [ju2 (Z0 1 a(1)sds; Z0 1 a(2)sdBs(2))j]  \n􀀌  \n􀀌  \n􀀔 􀀌  \n􀀌  \n􀀌  \nsup EP  \n􀀒2􀀂  \n\"u 1 ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2s[uA]p(0 ; 1) EP [u1 (Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \n+ 2cN  \n2 + sup EP 􀀒 2􀀂  \n\" 􀀌 Sn􀀒~~n~~ 􀀌 g + 􀀌 Sp􀀒~~nn~~ 􀀌 g \\# + a2s[uA]p(0 ; 1) EP 􀀔 􀀌 Z0 1 a(1)s ds 􀀌 g + 􀀌 Z0 1 a(2)s dBs(2) 􀀌 g 􀀕!  \nBy the Burkholder-Davis-Gundy inequality (Mao (2008, Theorem 1.7.3)),  \nlimnp 􀀌 s􀀒p􀀂 EP \"u ~~ ~~Sn􀀒~~n~~ ; Sp􀀒~~nn~~ !\\# 􀀀 a2 s[uA]p(0 ; 1) EP [u(Z0 1 a(1)sds; Z0 1 a(2)sdBs(2) )] 􀀌  \n􀀔 2cN 2 + maxfj~~􀀖~~ jg ; j􀀖 jgg + ~~􀀛~~g + sunp s􀀒p􀀂 EP \" 􀀌 Sn􀀒~~n~~ 􀀌 g + 􀀌 Sp􀀒~~nn~~ 􀀌 g \\#! :  \nSince N can be arbitrarily large, it su¢ ces to prove  \nsup  \nn  \nsup EP  \n􀀒2􀀂  \n\" 􀀌 Sn􀀒~~n~~ 􀀌 g + 􀀌 Sp􀀒~~nn~~ 􀀌 g \\# \u003C 1  \nStep 3: Prove the preceding in","cbCait9kx9FiL4SR","https://ap.wps.com/l/cbCait9kx9FiL4SR","pdf",152407,"English","# Lemma: extension of the CLT and Proposition 6\n## Perturbed returns and the case τ = 0\n## Definition of V_{τ,n} and PDE/HJB characterization\n# Generalization to continuous u with growth control\n## Step 1: compact support case\n## Step 2: polynomial growth via decomposition\n## Step 3: moment inequality and completion\n# Proof of Corollary 7","[{\"question\":\"What does the appendix prove about Proposition 6 and the CLT when τ = 0?\",\"answer\":\"It shows the CLT and Proposition 6 remain valid when τ = 0 by adding a small perturbation to the random returns of the K arms and controlling the resulting change in the objective via bounds derived from the associated PDE solution.\"},{\"question\":\"How is V_{τ,n} defined in this appendix?\",\"answer\":\"V_{τ,n} is defined through a supremum over squared terms involving expectations of u evaluated on the perturbed reward pair (Z^1_{τ,n}, Z^2_{τ,n}). It is then related to logarithmic asymptotics and the solution of a PDE with a modified function G_{τ}.\"},{\"question\":\"How does the appendix extend validity from compactly supported u to general continuous u?\",\"answer\":\"It proves the CLT first for u with compact support, then decomposes a general growth-controlled u into u1 (compact support) and u2 (vanishing outside a ball of radius N). It bounds the contribution of u2 using moment inequalities and the Burkholder–Davis–Gundy inequality to ensure the error can be made arbitrarily small.\"}]","Approximate optimality and the risk/reward tradeo§ given repeated gambles - Online appendix | PDF",15]