Matches in SemOpenAlex for { <https://semopenalex.org/work/W3048492742> ?p ?o ?g. }
Showing items 1 to 87 of
87
with 100 items per page.
- W3048492742 abstract "In this article, we try to give an answer to the simple question: ``textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?''}. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum $sum_{i=1}^{n}X_i$ in $mathcal{R}^p$ uniformly over the class of hyper-rectangles $mathcal{A}^{re}={prod_{j=1}^{p}[a_j,b_j]capmathcal{R}:-inftyleq a_jleq b_j leq infty, j=1,ldots,p}$, where $X_1,dots,X_n$ are independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the critical cut-off rate of $log p$ below which the uniform central limit theorem (CLT) holds and above which it fails. According to some recent results of Chernozukov et al. (2017), it is well known that the CLT holds uniformly over $mathcal{A}^{re}$ if $log p=obig(n^{1/7}big)$. They also conjectured that for CLT to hold uniformly over $mathcal{A}^{re}$, the optimal rate is $log p = obig(n^{1/3}big)$. We show instead that under some conditions, the CLT holds uniformly over $mathcal{A}^{re}$, when $log p=obig(n^{1/2}big)$. More precisely, we show that if $log p =epsilon sqrt{n}$ for some sufficiently small $epsilon>0$, the normal approximation is valid with an error $epsilon$, uniformly over $mathcal{A}^{re}$. Further, we show by an example that the uniform CLT over $mathcal{A}^{re}$ fails if $limsup_{trightarrow infty} n^{-(1/2+delta)} log p >0$ for some $delta>0$. Hence the critical rate of the growth of $p$ for the validity of the CLT is given by $log p=obig(n^{1/2}big)$." @default.
- W3048492742 created "2020-08-18" @default.
- W3048492742 creator A5010316541 @default.
- W3048492742 creator A5065965915 @default.
- W3048492742 date "2020-08-10" @default.
- W3048492742 modified "2023-09-27" @default.
- W3048492742 title "Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate" @default.
- W3048492742 cites W131372963 @default.
- W3048492742 cites W1509689762 @default.
- W3048492742 cites W1587704401 @default.
- W3048492742 cites W1627839092 @default.
- W3048492742 cites W1971847620 @default.
- W3048492742 cites W1975980877 @default.
- W3048492742 cites W1987673206 @default.
- W3048492742 cites W1993474005 @default.
- W3048492742 cites W2007597463 @default.
- W3048492742 cites W2032338935 @default.
- W3048492742 cites W2034681107 @default.
- W3048492742 cites W2066393385 @default.
- W3048492742 cites W2083760632 @default.
- W3048492742 cites W2100691045 @default.
- W3048492742 cites W2169419483 @default.
- W3048492742 cites W227597898 @default.
- W3048492742 cites W3029642367 @default.
- W3048492742 cites W3108539010 @default.
- W3048492742 cites W3125523732 @default.
- W3048492742 cites W393766269 @default.
- W3048492742 hasPublicationYear "2020" @default.
- W3048492742 type Work @default.
- W3048492742 sameAs 3048492742 @default.
- W3048492742 citedByCount "6" @default.
- W3048492742 countsByYear W30484927422020 @default.
- W3048492742 countsByYear W30484927422021 @default.
- W3048492742 crossrefType "posted-content" @default.
- W3048492742 hasAuthorship W3048492742A5010316541 @default.
- W3048492742 hasAuthorship W3048492742A5065965915 @default.
- W3048492742 hasConcept C105795698 @default.
- W3048492742 hasConcept C111472728 @default.
- W3048492742 hasConcept C114614502 @default.
- W3048492742 hasConcept C122123141 @default.
- W3048492742 hasConcept C134306372 @default.
- W3048492742 hasConcept C138885662 @default.
- W3048492742 hasConcept C141513077 @default.
- W3048492742 hasConcept C151201525 @default.
- W3048492742 hasConcept C166785042 @default.
- W3048492742 hasConcept C2780586882 @default.
- W3048492742 hasConcept C33676613 @default.
- W3048492742 hasConcept C33923547 @default.
- W3048492742 hasConceptScore W3048492742C105795698 @default.
- W3048492742 hasConceptScore W3048492742C111472728 @default.
- W3048492742 hasConceptScore W3048492742C114614502 @default.
- W3048492742 hasConceptScore W3048492742C122123141 @default.
- W3048492742 hasConceptScore W3048492742C134306372 @default.
- W3048492742 hasConceptScore W3048492742C138885662 @default.
- W3048492742 hasConceptScore W3048492742C141513077 @default.
- W3048492742 hasConceptScore W3048492742C151201525 @default.
- W3048492742 hasConceptScore W3048492742C166785042 @default.
- W3048492742 hasConceptScore W3048492742C2780586882 @default.
- W3048492742 hasConceptScore W3048492742C33676613 @default.
- W3048492742 hasConceptScore W3048492742C33923547 @default.
- W3048492742 hasLocation W30484927421 @default.
- W3048492742 hasOpenAccess W3048492742 @default.
- W3048492742 hasPrimaryLocation W30484927421 @default.
- W3048492742 hasRelatedWork W1485751852 @default.
- W3048492742 hasRelatedWork W1627839092 @default.
- W3048492742 hasRelatedWork W1976337900 @default.
- W3048492742 hasRelatedWork W1996048372 @default.
- W3048492742 hasRelatedWork W2006160689 @default.
- W3048492742 hasRelatedWork W2034398085 @default.
- W3048492742 hasRelatedWork W2114331905 @default.
- W3048492742 hasRelatedWork W2162021125 @default.
- W3048492742 hasRelatedWork W2329796247 @default.
- W3048492742 hasRelatedWork W2341248350 @default.
- W3048492742 hasRelatedWork W2491158392 @default.
- W3048492742 hasRelatedWork W2560249067 @default.
- W3048492742 hasRelatedWork W2741452392 @default.
- W3048492742 hasRelatedWork W2741595416 @default.
- W3048492742 hasRelatedWork W2748531157 @default.
- W3048492742 hasRelatedWork W2974878803 @default.
- W3048492742 hasRelatedWork W2988920802 @default.
- W3048492742 hasRelatedWork W3046651378 @default.
- W3048492742 hasRelatedWork W3170934313 @default.
- W3048492742 hasRelatedWork W3198113772 @default.
- W3048492742 isParatext "false" @default.
- W3048492742 isRetracted "false" @default.
- W3048492742 magId "3048492742" @default.
- W3048492742 workType "article" @default.