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values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
d8394b6502968e20c5aa3a5b09e98971861358f2 | subsection | 20 | 60 | The singular distributions | If k\ge 2 is even, then
X\in {F}_k^0 if and only if either
\operatorname{{P}}(X=\pm 1)={1}/{2}, or\operatorname{{P}}\mathopen {}\mathclose {\left(X=-\sqrt{{p_k}/{(1-p_k)}}\right)}=1-p_k,
\qquad \operatorname{{P}}\mathopen {}\mathclose {\left(X=\sqrt{{(1-p_k)}/{p_k}}\right)}=p_kand k\in \lbrace 4,6,\ldots \rbrace , or\o... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
0.0032053859904408455,
-0.008435828611254692,
-0.035787440836429596,
-0.037282396107912064,
-0.020898887887597084,
-0.02033446542918682,
0.06596115976572037,
0.014415656216442585,
-0.0007198294042609632,
0.015544501133263111,
-0.04362833499908447,
-0.011791854165494442,
-0.026985498145222664... | |
ab612a4f62fbb0708b71423c6ba2ab79fc2db443 | subsection | 21 | 60 | The singular distributions | Then, we get the equationp_k(t)\doteq t^k-k t^{k-1}-k t+1=0, \quad 0<t<\infty .Since for any t>0 we have p_k(1/t)=p_k(t)/t^k, it
follows that 1/t is a root of (REF ) whenever
t is; equivalently, 1-p is a root of (REF )
if p is. For odd k\ge 3, we see that p_k(0)=1>0,
p_k(1)=-2(k-1)<0 and p_k(\infty )=\infty . Thus,
(RE... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05008257180452347,
0.02327115274965763,
-0.035097476094961166,
-0.029863374307751656,
0.0005326614482328296,
-0.0015374220674857497,
0.07550535351037979,
0.019074715673923492,
-0.000865515205077827,
0.020112380385398865,
-0.02285913936793804,
-0.014565452933311462,
-0.05005205422639847,
... | |
a01889e4e3757aaf857b0023e8205674655940d6 | subsection | 22 | 60 | The singular distributions | However, the root p_k cannot lie in ({(k-2)}/{(k-1)},{k}/{(k+1)}]
because for all p in this interval the lhs of (REF )
is non-positive, while its rhs is strictly positive
(p>{(k-2)}/{(k-1)} implies (k+1)p-1>(k+1){(k-2)}/{(k-1)}-1
={[(k-3)(k+1)+2]}/{(k-1)}>0,
since k\ge 3). This verifies that p_k>{k}/{(k+1)}.
Finally, i... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.032431453466415405,
0.0015874402597546577,
-0.03767906129360199,
-0.04274361580610275,
-0.007318433374166489,
-0.017497118562459946,
0.04307921975851059,
0.016886932775378227,
-0.0006020832224749029,
0.011288464069366455,
-0.04908956214785576,
0.008519739843904972,
-0.01960226520895958,
... | |
2b4c5466ed1fbfc56545956ccce9fe26e32eb251 | subsection | 23 | 60 | The singular distributions | Therefore, in view of the assumption \textrm {\scalebox {.882}{Ė}}_k=0,
(REF ) simplifies toX \mathopen {}\mathclose {\left(X^{k-1}-k\textrm {\scalebox {.882}{Ė}}_{k-1}\right)}=0 \quad \textrm {with probability one.}It follows that the support of X is a subset
of A\doteq \lbrace -(k\textrm {\scalebox {.882}{Ė}}_{k-1})^... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03828930854797363,
0.01880905032157898,
-0.04832690581679344,
-0.03374340757727623,
-0.004732772242277861,
-0.00832144170999527,
0.07499212771654129,
0.04857097938656807,
0.037404537200927734,
-0.004351404495537281,
-0.04106566682457924,
-0.012600386515259743,
-0.018702268600463867,
-0.... | |
450d51879c398cda6ac3b17db703106814b8e71a | subsection | 24 | 60 | The singular distributions | On the other hand, it is easily seen that
for this value of a=\surd {k}, \textrm {\scalebox {.882}{Ė}}_k=0 and
\textrm {\scalebox {.882}{Ė}}_{k-1}=k^{(k-3)/2} so that
k\textrm {\scalebox {.882}{Ė}}_{k-1}=k^{(k-1)/2}=(\pm \surd {k})^{k-1}; hence,
A=\lbrace -\surd {k},0,\surd {k}\rbrace and
x(x^{k-1}-k\textrm {\scalebox ... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05013478547334671,
-0.00483649643138051,
-0.03564055263996124,
-0.05843464285135269,
-0.011740327812731266,
0.00521029531955719,
0.04625948518514633,
0.035945694893598557,
-0.011107158847153187,
0.0034709873143583536,
-0.026303216814994812,
0.004241470247507095,
-0.02456391043961048,
0.... | |
2f989f7d2f64cd897ffcd68468e33109d65ae29c | subsection | 25 | 60 | The singular distributions | Therefore, {F}_3^0 admits the parametrization{F}_3^0=\lbrace X_\textrm {\scalebox {.93}{Ē}}, \ -\surd {2}\le \textrm {\scalebox {.93}{Ē}}\le \surd {2}\rbrace ,where X_{\textrm {\scalebox {.93}{Ē}}} is characterized by\operatorname{{E}}\mathopen {}\mathclose {\left(X_{\textrm {\scalebox {.93}{Ē}}}\right)}=0, \quad \oper... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.04477137699723244,
0.00020087702432647347,
-0.03274688869714737,
-0.02488824725151062,
0.013107911683619022,
0.0021649417467415333,
0.02906934916973114,
0.04480189457535744,
-0.015129795297980309,
0.022965548560023308,
-0.041231174021959305,
0.0007901560747995973,
0.027329767122864723,
... | |
53326d3307883df4e7264f4dcde750df5be82449 | subsection | 26 | 60 | The singular distributions | The values \textrm {\scalebox {.93}{Ē}}=\pm 2 are impossible because
the equations y(y^2-3)=\pm 2 have exactly two real
solutions, say \textrm {\scalebox {.883}{ċ}},\textrm {\scalebox {.882}{Č}}, with |\textrm {\scalebox {.883}{ċ}}|=1 and
|\textrm {\scalebox {.882}{Č}}|=2, so that \operatorname{{E}}(X)=0 and \operatorn... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03891923651099205,
0.016147667542099953,
-0.010386094450950623,
-0.019688555970788002,
-0.006433120928704739,
-0.03141011297702789,
0.018269147723913193,
0.01889490894973278,
0.00976033415645361,
-0.0098595404997468,
-0.03327213227748871,
-0.0023637714330106974,
0.030387530103325844,
-0... | |
b1ac4525fd32f1b518fc9c36f0046e9a2bcf658f | subsection | 27 | 60 | The singular distributions | Then,
\lbrace y\colon y(y^2-3)=\textrm {\scalebox {.93}{Ē}}\rbrace =\lbrace -\textrm {\scalebox {.883}{ċ}},\textrm {\scalebox {.882}{Č}},\textrm {\scalebox {.883}{č}}\rbrace
where 0<\textrm {\scalebox {.882}{Č}}<1<\textrm {\scalebox {.883}{č}}<\surd {3}<\textrm {\scalebox {.883}{ċ}}
and, by definition, the numbers
\te... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05633607879281044,
-0.009819140657782555,
-0.016479676589369774,
-0.006927567534148693,
0.017929276451468468,
-0.022903697565197945,
0.03080783784389496,
0.00004392911432660185,
-0.014381568878889084,
0.040344689041376114,
-0.02156090922653675,
0.0033817668445408344,
-0.003419914282858371... | |
da9cb0b5e9ef9bb2a9b7339c008c3838783ad9f4 | subsection | 28 | 60 | The singular distributions | Also, from (REF ) we get the equation
3(\textrm {\scalebox {.883}{ċ}}+\textrm {\scalebox {.882}{Č}})=\textrm {\scalebox {.883}{ċ}}^3+\textrm {\scalebox {.882}{Č}}^3
=(\textrm {\scalebox {.883}{ċ}}+\textrm {\scalebox {.882}{Č}})(\textrm {\scalebox {.883}{ċ}}^2-\textrm {\scalebox {.883}{ċ}}\textrm {\scalebox {.882}{Č}}+\... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.053379401564598083,
0.0037463244516402483,
-0.05249432474374771,
-0.04941180720925331,
-0.01802203431725502,
0.00537151237949729,
0.03457910940051079,
-0.01292520109564066,
-0.01072013471275568,
0.009262806735932827,
-0.011582323350012302,
0.006042951252311468,
-0.01414599921554327,
0.0... | |
462d4f8cde11a66d5a8b9f5bb6881d4e0cce00e5 | subsection | 29 | 60 | The singular distributions | Since \textrm {\scalebox {.883}{č}}>0 it follows that\textrm {\scalebox {.883}{č}}=\textrm {\scalebox {.883}{č}}(\textrm {\scalebox {.882}{Č}})=\frac{1}{2}(-\textrm {\scalebox {.882}{Č}}+\textrm {\scalebox {.882}{Ď}}),
\quad \textrm {where}
\ \textrm {\scalebox {.882}{Ď}}=\textrm {\scalebox {.882}{Ď}}(\textrm {\scalebo... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05810815468430519,
-0.004917372018098831,
-0.06061071157455444,
-0.015404459089040756,
-0.002981323515996337,
-0.008011233992874622,
0.04776221513748169,
-0.013176573440432549,
-0.028581032529473305,
-0.004207805264741182,
-0.02507134899497032,
-0.020203569903969765,
0.010567199438810349,... | |
0fed47647a150913075ecf644864453591240d15 | subsection | 30 | 60 | The singular distributions | Since \operatorname{{P}}(X\in \lbrace -\textrm {\scalebox {.883}{ċ}},\textrm {\scalebox {.882}{Č}},\textrm {\scalebox {.883}{č}}\rbrace )=1
and \operatorname{{E}}(X)=0, \operatorname{{E}}(X^2)=1, we get the system of
equations (in p_1, p_2, p_3)p_1+ p_2 + p_3=1,
\quad -\textrm {\scalebox {.883}{ċ}}p_1+\textrm {\scalebo... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0386439748108387,
-0.008760522119700909,
-0.06245305761694908,
-0.028509851545095444,
0.0011198663851246238,
-0.024770604446530342,
0.05775228887796402,
0.009103922173380852,
-0.019154103472828865,
-0.001567240571603179,
-0.034828417003154755,
-0.02941032312810421,
-0.0014613588573411107,... | |
f01d4d7bdbfceacc29e2525726978221c9cb3e7f | subsection | 31 | 60 | The singular distributions | Now, from \textrm {\scalebox {.882}{Ė}}_3=\textrm {\scalebox {.93}{Ē}}=\textrm {\scalebox {.883}{č}}(\textrm {\scalebox {.883}{č}}^2-3) and the fact
that \textrm {\scalebox {.883}{č}}\in [\surd {2},\surd {3}), we conclude that all
possible values of \textrm {\scalebox {.93}{Ē}} (with \textrm {\scalebox {.93}{Ē}}<0) are... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.07261133193969727,
0.004526767414063215,
-0.02503260411322117,
-0.00649840896949172,
-0.011395097710192204,
-0.0006692903698422015,
0.018335886299610138,
0.0030222940258681774,
-0.00939676072448492,
0.0065823085606098175,
-0.008008603006601334,
0.0046526165679097176,
-0.018381649628281593... | |
64f13da1b243e92501c8245201987855c38bb04b | subsection | 32 | 60 | The singular distributions | On the other hand, each value of
\textrm {\scalebox {.883}{č}}\in (\surd {2},\surd {3}) corresponds to
a unique value of
\textrm {\scalebox {.882}{Ė}}_3=\textrm {\scalebox {.93}{Ē}}=\textrm {\scalebox {.883}{č}}(\textrm {\scalebox {.883}{č}}^2-3)\in (-\surd {2},0),
which, in turn, uniquely determines
\textrm {\scalebox... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0529039092361927,
-0.0160633847117424,
-0.03340817615389824,
-0.01896180957555771,
-0.009252081625163555,
-0.017421068623661995,
0.032340336591005325,
0.00858849473297596,
-0.02163141407072544,
0.026955364271998405,
-0.033042062073946,
-0.024056939408183098,
0.0007574998890049756,
-0.01... | |
b7e0bed673a3b76504e58edf5a996fc03fdcdb54 | subsection | 33 | 60 | The singular distributions | By the previous arguments it follows
that, necessarily, -\surd {2}\le -\textrm {\scalebox {.93}{Ē}}<0, that -X is determined
by the value of -\textrm {\scalebox {.93}{Ē}}, and that -X is a two-valued
random variable, if -\textrm {\scalebox {.93}{Ē}}=-\surd {2}, and a three-valued
random variable otherwise; thus, the sa... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.034148748964071274,
0.012199251912534237,
-0.026412639766931534,
-0.01773049496114254,
-0.007934472523629665,
-0.004314369522035122,
0.03720046952366829,
0.0397639162838459,
-0.015586662106215954,
0.01776101253926754,
0.005706717260181904,
-0.022384367883205414,
-0.0200803205370903,
0.0... | |
998b8c5868bf5b3263fb506778b659b2b9109619 | subsection | 34 | 60 | Limiting distribution under singularness | If the random sample comes from a singular distribution of order k\ge 2,
then the asymptotic normality of (REF ) reduces to
\surd {n}(M_{k,n}-\textrm {\scalebox {.882}{Ė}}_k){\mathrm {p}}0
(see def.singular).
This shows that the order of
convergence of M_{k,n} to \textrm {\scalebox {.882}{Ė}}_k
is faster than o(1/\surd... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.09430956095457077,
0.01149111706763506,
-0.002863241359591484,
-0.0300783421844244,
-0.01849566213786602,
0.008103297092020512,
-0.00813381839543581,
0.013635209761559963,
0.023058535531163216,
0.027697712182998657,
-0.03216902166604996,
-0.0200827494263649,
-0.028125004842877388,
0.022... | |
76a5084e5991c8aa1d52b7bffe054159a16523a4 | subsection | 35 | 60 | Limiting distribution under singularness | Therefore, the Taylor expansion suggests the approximation
n[g(Xn)-g(Ė)]
=
n[g(Ė)]' (Xn-Ė)
+12[n(Xn-Ė)]'
Hk(Ė)[n(Xn-Ė)]
+op(1)
and, by (REF ), the rhs of the
above equals to\frac{1}{2}[\surd {n}({X}_n-{\textrm {\scalebox {.882}{Ė}}})]^{\prime }
\mathbf {H}_k({\textrm {\scalebox {.882}{Ė}}})[\surd {n}({X}_n-{\textrm {... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.047776419669389725,
0.001739941886626184,
-0.0075508709996938705,
-0.032613661140203476,
-0.018762771040201187,
0.0024921686854213476,
-0.0004030468116980046,
0.013675466179847717,
-0.0233695637434721,
0.019800061360001564,
-0.01615428738296032,
0.02158481255173683,
-0.05509847775101662,
... | |
1f15b757b0ac3127d806be3dd0dae56cdded234c | subsection | 36 | 60 | Limiting distribution under singularness | Indeed,
observing that \textrm {\scalebox {.883}{ě}}^2(\textrm {\scalebox {.882}{Ė}}_{2k-2}-\textrm {\scalebox {.882}{Ė}}_{k-1}^2)-\textrm {\scalebox {.882}{Ė}}_k^2
=\operatorname{\mathsf {Var}}[\textrm {\scalebox {.883}{ě}}(X-\textrm {\scalebox {.882}{Ė}})^{k-1}-\textrm {\scalebox {.882}{Ė}}_k {(X-\textrm {\scalebox {... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.04004044085741043,
0.006954432465136051,
-0.013634196482598782,
-0.02534572035074234,
-0.0029793810099363327,
0.0043794228695333,
0.03305167332291603,
0.020554296672344208,
-0.003914013970643282,
0.0212104469537735,
-0.02884010411798954,
0.00010210627078777179,
-0.029053732752799988,
0.... | |
e3d02b904b2a1ad7044f6ba94fb0ec1b9b7ca058 | subsection | 37 | 60 | Limiting distribution under singularness | If Z_1, Z_2 are
independent and identically distributed standard
normal random variables, then, for arbitrary constants
\textrm {\scalebox {.883}{ċ}},\textrm {\scalebox {.882}{Č}}\in {R},\textrm {\scalebox {.883}{ċ}}Z_1^2+\textrm {\scalebox {.882}{Č}}Z_1Z_2 {\mathrm {d}}\frac{1}{2}\mathopen {}\mathclose {\left(\sqrt{\t... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03384421020746231,
-0.00975805800408125,
-0.028671449050307274,
-0.006626171991229057,
-0.002138151554390788,
-0.01344307977706194,
0.017120471224188805,
-0.013763516210019588,
0.005878486204892397,
-0.010627814568579197,
-0.047027889639139175,
-0.014618013985455036,
0.008987484499812126,... | |
8c395e40b5c09be107cf080e8e6b7baa283fd960 | subsection | 38 | 60 | Limiting distribution under singularness | Also, the moment generating function of
the lhs of (REF ) isM_1(t)=\operatorname{{E}}\mathopen {}\mathclose {\left[\exp \mathopen {}\mathclose {\left(\textrm {\scalebox {.883}{ċ}}t Z_1^2 +\textrm {\scalebox {.882}{Č}}t Z_1 Z_2\right)}\right]}=\frac{1}{2\textrm {\scalebox {.883}{ę}}}
\iint _{{R}^2}
e^{-\frac{1}{2}\textr... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05351211503148079,
0.0031843369361013174,
-0.03249167650938034,
-0.006711895577609539,
-0.004538156557828188,
-0.003636245848610997,
-0.005075871013104916,
0.029639121145009995,
0.0019525514217093587,
-0.001636978005990386,
-0.03737305477261543,
-0.007177152205258608,
-0.02606961317360401... | |
3e570ed81542352911ab36a441cf9a8c6b805c40 | subsection | 39 | 60 | Limiting distribution under singularness | If M_{k,n} is the sample central moment of a singular
distribution of order k\ge 2, thenn(\textrm {\scalebox {.882}{Ė}}_k-M_{k,n}){\mathrm {d}}\frac{k}{2}\mathopen {}\mathclose {\left(\textrm {\scalebox {.883}{ě}}\surd {\textrm {\scalebox {.93}{Ē}}_k}+\textrm {\scalebox {.883}{ċ}}_k\right)}
Z_1^2
- \frac{k}{2}\mathopen... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.038302648812532425,
-0.01741168275475502,
-0.01187992561608553,
-0.02823103591799736,
-0.02043316699564457,
0.014168928377330303,
0.009621442295610905,
0.0015841807471588254,
-0.021180907264351845,
-0.001378170563839376,
-0.02012796513736248,
-0.01171206496655941,
-0.03744808956980705,
... | |
b0d77734f36c4c3be4aa81e7f502d778cb6c3905 | subsection | 40 | 60 | Limiting distribution under singularness | \end{split}If M_{k,n} is the sample central moment of a singular
distribution of order k\ge 2, then
there exists a constant \textrm {\scalebox {.883}{ĕ}}_k\in {R}
such thatn(\textrm {\scalebox {.882}{Ė}}_k-M_{k,n}){\mathrm {d}}\textrm {\scalebox {.883}{ĕ}}_k \textrm {\scalebox {.883}{ğ}}_1^2if and only if\textrm {\scal... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05558545142412186,
-0.028662679716944695,
-0.007123698480427265,
-0.04191039130091667,
-0.004013996105641127,
-0.00209284876473248,
-0.004800006747245789,
0.039071597158908844,
-0.017979038879275322,
0.014438175596296787,
-0.008386657573282719,
-0.025198128074407578,
-0.04215458780527115,... | |
43e31a1a0612916b364780a182d64e99c014dbe1 | subsection | 41 | 60 | Limiting distribution under singularness | In particular,
from (REF ) we can show thatn(\textrm {\scalebox {.882}{Ė}}_k-M_{k,n}){\mathrm {d}}\frac{k(k-1)}{2} p_k^{k-1}
\frac{(k+1)p_k^2-(k+1)p_k+1}{(k+1)p_k-1}\textrm {\scalebox {.883}{ğ}}_1^2, \quad k=3,4,\ldots ~.For example, the two-valued standardized distribution
of cor.even
with p_6={1}/{2}+\sqrt{15(4\surd ... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0528230294585228,
-0.0028417936991900206,
-0.0022505479864776134,
-0.07183443754911423,
-0.011412948369979858,
0.03219427913427353,
-0.00408531678840518,
0.009200545027852058,
-0.0036237637978047132,
0.008941160514950752,
-0.03939603269100189,
0.03179757297039032,
-0.04501095786690712,
... | |
d758c1c4372d73e33cbc6079236440811a05320a | subsection | 42 | 60 | Proofs | We shall make use of the following Lemmas.
For the proof of lem.ui see, e.g., ;
for more general results, see .If X,X_1,\ldots ,X_n are independent and identically distributed
with \operatorname{{E}}(X)=\textrm {\scalebox {.882}{Ė}}, \operatorname{\mathsf {Var}}(X)=\textrm {\scalebox {.883}{ě}}^2
and \operatorname{{E}}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 111,
"openalex_id": "https://openalex.org/W1592406844",
"raw": "Gut, A. (1988). Stopped Random Walks: Limit Theorems and Applications. Springer–Verlag, N.Y.",
"source_ref_id": "b1dbe1aafa4530a9f1280345d23400390d94a226",
... | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0699995756149292,
0.026410039514303207,
-0.01925445906817913,
-0.014936700463294983,
-0.02033771388232708,
0.003503411542624235,
-0.004279616288840771,
0.031002426519989967,
0.002498350106179714,
0.005042471457272768,
-0.038630977272987366,
0.009123745374381542,
-0.023129763081669807,
0... | |
50e209f8b2a42567b9e2bf901924def858f14e1e | subsection | 43 | 60 | Proofs | If j<\textrm {\scalebox {.883}{ė}} (and thus, \textrm {\scalebox {.883}{ė}}\ge 3), we apply
the inequality\mathopen {}\mathclose {\left|\sum _{i=1}^{n} x_i\right|}^p
\le \mathopen {}\mathclose {\left(\sum _{i=1}^{n} |x_i|\right)}^p
\le n^{p-1} \sum _{i=1}^n
|x_i|^p,
\quad p>1,(the last inequality is a by-product of Höl... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05995325371623039,
0.009313348680734634,
-0.039328113198280334,
-0.020075950771570206,
-0.012158459052443504,
0.03798564895987511,
0.009885421954095364,
0.032707322388887405,
-0.018352102488279343,
0.0038748413790017366,
-0.027413740754127502,
-0.00025123541126959026,
-0.01320344582200050... | |
4fb2b1ca078c0e44bc8410ff89157d88631c87f1 | subsection | 44 | 60 | Proofs | \qquad \mathrm {}Proof of prop.exp.cov.conv
(REF )
Observe that the statement in prop.exp.cov.conv(REF ) is equivalent to\operatorname{{E}}[\surd {n}(M_{k,n}-\textrm {\scalebox {.882}{Ė}}_k)]\rightarrow 0.WritingM_{k,n}-\textrm {\scalebox {.882}{Ė}}_k=
(m_{k,n}-\textrm {\scalebox {.882}{Ė}}_k)+(-1)^{k-1}(k-1) m_{1,n}^... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.06229057163000107,
0.02226407267153263,
0.005314231384545565,
-0.059818483889102936,
-0.015572642907500267,
0.02914625033736229,
-0.031435221433639526,
0.054141830652952194,
0.007927474565804005,
0.025056619197130203,
-0.015748130157589912,
-0.01057505328208208,
-0.05929964780807495,
0.... | |
4b353c1f829435299d9af26fdc8e4562d1bbda2d | subsection | 45 | 60 | Proofs | To show (iii), we assume that k\ge 4 and 2\le j\le k-2, and
we use Hölder's inequality with p=k/(k-j)>1, lem.holder
with \textrm {\scalebox {.883}{ė}}=k and lem.ui with \textrm {\scalebox {.883}{ċ}}=\textrm {\scalebox {.882}{Ď}}=k
to obtain
|nE(m1,nk-jmj,n)|
nE(|m1,n|k-j |mj,n|)n(E|m1,n|k)(k-j)/k
(E|mj,n|k/j)j/k
n
[n... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.06590183079242706,
0.0262234378606081,
-0.03124235011637211,
-0.025872571393847466,
-0.003565868828445673,
0.051989223808050156,
-0.002593359211459756,
0.011754019185900688,
0.0007432071724906564,
0.009419995360076427,
-0.04680250585079193,
-0.004286670126020908,
-0.019007796421647072,
... | |
86cfa208dc37a8745ce12fe3849621d8c0ed6945 | subsection | 46 | 60 | Proofs | Since \textrm {\scalebox {.882}{Ė}}_1=0, (REF ) is satisfied for k=2.If k=3,
n\operatorname{{E}}[m_{1,n}(M_{3,n}-\textrm {\scalebox {.882}{Ė}}_3)]=n\operatorname{{E}}[(\bar{X}_n-\textrm {\scalebox {.882}{Ė}})(m_{3,n}-\textrm {\scalebox {.882}{Ė}}_3)]
+2n\operatorname{{E}}(\bar{X}_n-\textrm {\scalebox {.882}{Ė}})^4-3n\l... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05280178040266037,
-0.00374266947619617,
-0.022036349400877953,
-0.035221535712480545,
0.014436556026339531,
0.00270876195281744,
0.023638715967535973,
0.022921467199921608,
-0.025134257972240448,
0.018572187051177025,
-0.020174553617835045,
-0.005490011535584927,
-0.018175410106778145,
... | |
e56adedf7dd8e0fe6c0501298b35556760b50bd6 | subsection | 47 | 60 | Proofs | Finally, -3n\operatorname{{E}}[m_{2,n}(\bar{X}_n-\textrm {\scalebox {.882}{Ė}})^2]=-3[\textrm {\scalebox {.882}{Ė}}_4+(n-1)\textrm {\scalebox {.882}{Ė}}_2^2]/n\rightarrow -3
\textrm {\scalebox {.882}{Ė}}_2^2=-3\textrm {\scalebox {.883}{ě}}^4, which verifies (REF ) for
k=3.In the general case when k\ge 4, we write M_{k,... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.020854931324720383,
0.0036557153798639774,
-0.03176296129822731,
-0.044577986001968384,
0.012395487166941166,
-0.005850669927895069,
0.022761929780244827,
0.020107388496398926,
-0.03334958106279373,
0.01646120660007,
-0.022822953760623932,
-0.007746226154267788,
-0.019695475697517395,
0... | |
658c8ad78b1e0b5ac66f5180f768a8cdefae56c2 | subsection | 48 | 60 | Proofs | To this end, we use Hölder's
inequality with p=(k+1)/(k+1-j)>1
and lem.holder with \textrm {\scalebox {.883}{ė}}=k+1
to obtain
|n E(m1,nk+1-jmj,n)|
n
E|m1,n|k+1-j |mj,n|
n (E|m1,n|k+1)(k+1-j)/(k+1)
(E|mj,n|(k+1)/j)j/(k+1)
n
[n-(k+1)/2E|n(Xn-Ė)|k+1](k+1-j)/(k+1)
(E|X-Ė|k+1)j/(k+1)
=
n-(k-1-j)/2
[E|n(Xn-Ė)|k+1](k+1... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.048695072531700134,
0.03566700592637062,
-0.016155414283275604,
-0.02028961293399334,
-0.0017705739010125399,
0.04524736478924751,
-0.003623145166784525,
0.01996925100684166,
0.00405792286619544,
-0.005968654993921518,
-0.044057447463274,
-0.006369107868522406,
-0.04652881249785423,
0.0... | |
44be6404eac2c3d5974016ac83fca1d887a08dad | subsection | 49 | 60 | Proofs | From (REF ), we have
Mr,n-Ėr
=
(mr,n-Ėr)-rm1,nmr-1,n
+j1=2r-2 (-1)r-j1 rj1 m1,nr-j1 mj1,n
+(-1)r-1(r-1) m1,nr,
Mk,n-Ėk
=
(mk,n-Ėk)-km1,nmk-1,n
+j2=2k-2 (-1)k-j2 kj2 m1,nk-j2 mj2,n
+(-1)k-1(k-1) m1,nk.
We shall show that the asymptotic covariance in (REF )
can be determined by using only the first two terms in
() ... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05607613921165466,
0.02677193284034729,
0.009724847972393036,
0.02291250415146351,
-0.007097231689840555,
-0.03151613473892212,
0.03447553887963295,
0.02524646744132042,
-0.00581965409219265,
0.01726827770471573,
0.0046069081872701645,
0.010266387835144997,
-0.07779878377914429,
0.00787... | |
9afb632161c87c4c6b86eafe9f2ad137bab1b09c | subsection | 50 | 60 | Proofs | Indeed, it is
easily seen that
() holds true if it can be shown thatn \operatorname{{E}}[(m_{r,n}-\textrm {\scalebox {.882}{Ė}}_r)(m_{k,n}-\textrm {\scalebox {.882}{Ė}}_k)]=\textrm {\scalebox {.882}{Ė}}_{r+k}-\textrm {\scalebox {.882}{Ė}}_r\textrm {\scalebox {.882}{Ė}}_k,
n\operatorname{{E}}[m_{1,n}m_{k-1,n}(m_{r,n}-\... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
0.0009167558746412396,
0.012959486804902554,
-0.03432471305131912,
0.018184470012784004,
-0.009115118533372879,
-0.031273629516363144,
0.015911411494016647,
0.05806215852499008,
-0.005850456655025482,
0.020625339820981026,
0.003274196293205023,
-0.027734369039535522,
-0.03484340012073517,
... | |
9cc20325fbb0ae8ab7156f4d07f7f093c4b9527e | subsection | 51 | 60 | Proofs | Since \operatorname{{E}}(m_{r,n})=\textrm {\scalebox {.882}{Ė}}_r and \operatorname{{E}}(m_{k,n})=\textrm {\scalebox {.882}{Ė}}_k,
we have
nE[(mr,n-Ėr)(mk,n-Ėk)]
=
n[E(mr,nmk,n)-ĖrĖk]
=
n{
1n2i1=1ni2=1n
E[(Xi1-Ė)r(Xi2-Ė)k]-ĖrĖk}
=
n{1n2[nĖr+k+n(n-1)ĖrĖk]-ĖrĖk}
=Ėr+k-ĖrĖk,
which shows (i). Also, (ii), (iii) and (i... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0720076635479927,
0.008795004338026047,
-0.030481209978461266,
-0.01942071132361889,
0.005190806929022074,
-0.0031369864009320736,
0.005221318453550339,
0.04692702740430832,
-0.004912387114018202,
0.000661246245726943,
-0.057606130838394165,
-0.02802501618862152,
-0.04619474709033966,
-... | |
8749e9a1d94e75a6a661d0c94cbbd25293ac4e20 | subsection | 52 | 60 | Proofs | Also, using Hölder's
inequality with p=(r+k)/(r+k-j_2)>1,
we obtain (xv) as follows:
|n E(m1,nr+k-j2mj2,n)|
n E(|m1,n|r+k-j2 |mj2,n|)
n (E|m1,n|r+k)r+k-j2r+k
(E|mj2,n|r+kj2)j2r+k
n[n-(r+k)/2E|n(Xn-Ė)|r+k]r+k-j2r+k
(E|X-Ė|r+k)j2r+k
=n-(r+k-j2-2)/2
[E|n(Xn-Ė)|r+k]r+k-j2r+k
(E|X-Ė|r+k)j2r+k
0,
because n^{-(r+k-j_2-2)/2}... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.044151872396469116,
0.035364214330911636,
-0.02982616052031517,
0.005251998081803322,
-0.012372898869216442,
0.03377755358815193,
0.009344513528048992,
0.024944130331277847,
0.010534508153796196,
0.0034383994061499834,
-0.05068159103393555,
-0.02299131639301777,
-0.05178004503250122,
-0... | |
23779f9da3c8395a9e2b6f98a415def18725ceb5 | subsection | 53 | 60 | Proofs | Regarding (xi), we have
|nE(m1,nr+k-j1-j2mj1,nmj2,n)|
nE(|m1,n|r+k-j1-j2 |mj1,nmj2,n|)
n (E|m1,n|r+k)r+k-j1-j2r+k
(E|mj1,nmj2,n|r+kj1+j2)j1+j2r+k
n (E|m1,n|r+k)r+k-j1-j2r+k
[(E|mj1,n|r+kj1)j1j1+j2
(E|mj2,n|r+kj2)j2j1+j2]j1+j2r+k
n (E|m1,n|r+k)r+k-j1-j2r+k
(E|X-Ė|r+k)j1+j2r+k
=
n-(r+k-j1-j2-2)/2 [E|n(Xn-Ė)|r+k]r+k-j1-j... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.02689545229077339,
0.02800910361111164,
-0.013371448032557964,
0.0074980794452130795,
-0.009534689597785473,
-0.0018087305361405015,
0.022883255034685135,
0.05769631266593933,
0.0035907640121877193,
0.007635379210114479,
0.0013548793504014611,
-0.054828278720378876,
-0.06025923788547516,
... | |
aa900e1bc94938f9a330d60ac6703d6b2c08ffa1 | subsection | 54 | 60 | Proofs | For the first quantity, we have
nE(|m1,n|k-j2|mj2,nmr,n|)
n (E|m1,n|r+k)k-j2r+k
(E|mj2,nmr,n|r+kr+j2)r+j2r+k
n (E|m1,n|r+k)k-j2r+k
[(E|mj2,n|r+kj2)j2r+j2
(E|mr,n|r+kr)rr+j2]r+j2r+k
n-(k-j2-2)/2 [E|n(Xn-Ė)|r+k]k-j2r+k
(E|X-Ė|r+k)r+j2r+k 0,
because k-j_2-2>0. Similarly, for the second quantity we
have
nE(|m1,n|k-j2|mj... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.07780656963586807,
0.03450950235128403,
-0.006132988259196281,
-0.005236687138676643,
-0.00018772264593280852,
0.012868596240878105,
0.00405051838606596,
0.04378526657819748,
-0.014050951227545738,
0.022563904523849487,
-0.016141049563884735,
-0.02473028376698494,
-0.06926310062408447,
... | |
8404cb12df55933653f822f09dc4109e1fc1ab52 | subsection | 55 | 60 | Proofs | However, since j_2=k-2 we haven\operatorname{{E}}\mathopen {}\mathclose {\left[m_{1,n}^{k-j_2}m_{j_2,n}(m_{r,n}-\textrm {\scalebox {.882}{Ė}}_r)\right]}
=n\operatorname{{E}}\mathopen {}\mathclose {\left(m_{1,n}^2 m_{r,n} m_{k-2,n}\right)}
-n\textrm {\scalebox {.882}{Ė}}_r \operatorname{{E}}\mathopen {}\mathclose {\left... | {
"cite_spans": []
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"Nickos Papadatos",
"Violetta Piperigou"
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] | 2,018 | en | Mathematics | [
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eed8c127be266276f1eb655a184f5bd4519cd102 | subsection | 56 | 60 | Proofs | Also, \surd {n}({m}_n-{\textrm {\scalebox {.882}{Ė}}}_{k}){\mathrm {d}}{W}_{k},
where {W}_{k}=(W_1,\ldots ,W_k)^{\prime }\sim N({0}_k,\operatorname{\mathrel {}_k), see (\ref {clt}).
Hence, {lem.delta2}
applies to {X}_n={m}_{k,n}, provided (\ref {sing.var})
is fulfilled for {m}_{k,n}, i.e., provided that
n[\nabla g_{k,k... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
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1d1cc3c8c95170de04b49ae2288b0c709dee818a | subsection | 57 | 60 | Proofs | Also,
\begin{}
\operatorname{\mathsf {Var}}[-k\textrm {\scalebox {.882}{Ė}}_{k-1}m_{1,n}+(m_{k,n}-\textrm {\scalebox {.882}{Ė}}_k)]
=
{
k^2 \textrm {\scalebox {.882}{Ė}}_{k-1}^2
\operatorname{\mathsf {Var}}(m_{1,n}) + \operatorname{\mathsf {Var}}(m_{k,n})-2k\textrm {\scalebox {.882}{Ė}}_{k-1}\mathsf {Cov}(m_{1,n},m_{k,... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
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7d75dc6f04088b9426c7137f57c329539fbb3814 | subsection | 58 | 60 | Proofs | Therefore, [\nabla g_{k,k}({\textrm {\scalebox {.882}{Ė}}}_{k})]^{\prime }({m}_{k,n}-{\textrm {\scalebox {.882}{Ė}}}_{k})=0
with probability one and, thus,
n[\nabla g_{k,k}({\textrm {\scalebox {.882}{Ė}}}_{k})]^{\prime }({m}_{k,n}-{\textrm {\scalebox {.882}{Ė}}}_{k}){\mathrm {p}}0
in a trivial sense. Now, a simple calc... | {
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} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
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3fa383557f99498a8503a0c3994937bd5ba3e26b | subsection | 59 | 60 | Proofs | Applying (\ref {delta2}), we see that
n(M_{k,n}-\textrm {\scalebox {.882}{Ė}}_k) converges weakly to the
distribution of
\frac{1}{2}{W}^{\prime }_k\mathbf {H}_k({\textrm {\scalebox {.882}{Ė}}}_k){W}_k
=\frac{1}{2}k(k-1)\textrm {\scalebox {.882}{Ė}}_{k-2} W_1^2 - k W_1 W_{k-1},
while, by (\ref {clt}), the distribution o... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
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fea7e04791db56b0d0d621a0c2857ed648935c2a | abstract | 0 | 18 | Abstract | Traffic control management at intersections, a challenging and complex field
of study, aims to attain a balance between safety and efficient traffic
control. Nowadays, traffic control at intersections is typically done by
traffic light systems which are not optimal and exhibit several drawbacks, e.g.
poor efficiency an... | {
"cite_spans": []
} | 1809.07829 | Personal Virtual Traffic Light Systems | [
"Vanessa Martins",
"João Rufino",
"Bruno Fernandes",
"Luís Silva",
"João Almeida",
"Joaquim Ferreira",
"José Fonseca"
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100f2211d7e32331340ce992e7c2f5f5edf4d65a | subsection | 1 | 18 | Introduction | Transportation has always been a fundamental component of the economic and social interactions of the human society; without it, the movement of people, animals and goods from one geographical location to another would be highly constrained. From domesticated animals and wheel carts, to wheeled motor vehicles and airpl... | {
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ae7a01d3d9a3f27dce935092e9ccc1330bb5906e | subsection | 2 | 18 | Introduction | A significant number of these solutions are based on future vehicular communication systems and do not address the integration of legacy systems, systems which, are expected to coexist for a significant amount of time and thus, should not be ignored.In this work, a solution based on an alternative technology to vehicul... | {
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"raw": "James Chang, “Market Penetration Analysis for VSC- A Safety Benefit Opportunities Estimation.” [Online]. Available: https://www.pcb.its.dot.gov/connected_vehicle/documents/2010/07/Market%20Penetrati... | 1809.07829 | Personal Virtual Traffic Light Systems | [
"Vanessa Martins",
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261db261df077207471b7e51f3ac73c463f723fb | subsection | 3 | 18 | Related work | As previously discussed, there have been efforts in the literature to improve traffic control management at intersections. The most relevant for this work are going to be briefly discussed in this section.Michel et al. propose a virtual traffic light management framework that dynamically optimizes traffic flows in road... | {
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"Vanessa Martins",
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a27a79819bfc91db4233b8b8bb5f15b7d645ad77 | subsection | 4 | 18 | Traffic Light Systems | In order to propose a new architecture, we should first study the main characteristics of the conventional traffic control systems (TCS). Typically, there are four different roles in any TCS: users, data collectors, indication providers and controllers. A user is any actor receiving a traffic signal service such as, fo... | {
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"Vanessa Martins",
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"Bruno Fernandes",
"Luís Silva",
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63e6642341d1be2e5ec1f11bae06e63e316b0dac | subsection | 5 | 18 | Proposed Architecture | As discussed in section , vehicles are one of the multiple actors in the complex road safety system. Despite public and private efforts, there is still a long time to go until autonomous and cooperative vehicles make the use of nowadays physical traffic lights obsolete. In these futuristic systems, there is an extreme ... | {
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4d90fcfb733e0a2f33301883fb02f0744df9fe89 | subsection | 6 | 18 | Users | In our proposal, an user is any sort of actor receiving information on the approaching intersection. Ideally, in order to improve the overall decision-making process, users are obliged to provide their own location. This is a deciding factor since it excludes the need for a vehicle and pedestrian detection. Clearly, th... | {
"cite_spans": []
} | 1809.07829 | Personal Virtual Traffic Light Systems | [
"Vanessa Martins",
"João Rufino",
"Bruno Fernandes",
"Luís Silva",
"João Almeida",
"Joaquim Ferreira",
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d4ab67d06c4380813892df6d851954524d7c8093 | subsection | 7 | 18 | Environment Perception | Regarding environment perception, controllers are a meeting point for heterogeneous data sources, ranging from vehicular communications to multiple wireless sensor networks. Therefore, controllers have a global view of the environment and are able to devise intelligent management strategies. Even though there is redund... | {
"cite_spans": []
} | 1809.07829 | Personal Virtual Traffic Light Systems | [
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"Bruno Fernandes",
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772f4afcc460996338fa393d786f244228bb2565 | subsection | 8 | 18 | Signaling | Smart devices have become workaday tools, commonly used in the almost everyday situation. In fact, we believe these devices can enhance or even replace the cumbersome, currently available, physical light systems. Their capabilities are unique since they provide direct communication with the users in the most comfortabl... | {
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} | 1809.07829 | Personal Virtual Traffic Light Systems | [
"Vanessa Martins",
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"João Almeida",
"Joaquim Ferreira",
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c0f4c6ba7f628ec28d61b6756be5771e99b02187 | subsection | 9 | 18 | Traffic control | The controller is responsible for the proper functioning of the system. Together with the information gathering system, controllers combine historical data and exploit the rich information gathered in real-time from multiple sources, e.g. pressure sensors and traffic cameras, to efficiently adapt the traffic control st... | {
"cite_spans": []
} | 1809.07829 | Personal Virtual Traffic Light Systems | [
"Vanessa Martins",
"João Rufino",
"Bruno Fernandes",
"Luís Silva",
"João Almeida",
"Joaquim Ferreira",
"José Fonseca"
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151015b27c89a8bb2d9bc34fed824c0d027e4b69 | subsection | 10 | 18 | Communication | There are a few requirements that come with the usage of smart devices. Firstly, the power consumption has to be low in order to reduce the battery drainage and secondly, it has to be compatible with most mobile phones, tablets, watches and other smart solutions. This confines us to a reduced array of communication tec... | {
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429611e0cb9e8273a841face7e013d06c533c7e3 | subsection | 11 | 18 | Implementation | In order to evaluate the proposed system, we implemented the solution using BLE as the communication technology. This technology was chosen due to the solution requirements of low power, small size and low cost of modules and the compatibility with most mobile phones, tablets and other smart devices. Raspberry PIs mod... | {
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a75f8ada251e80fe9f0e140ac10c2939ef556cd5 | subsection | 12 | 18 | Controller | The controller entity is responsible for defining all the temporal states (Table REF ). In this case the phases are statically defined and therefore, the controller continuously broadcasts the same information to all retransmitters and/or users. The controllers hardware is composed by a Module RN3020 connected to a Ra... | {
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099ce8ec20f03d53aeca43c6ebd58fff8c6564d8 | subsection | 13 | 18 | Retransmitter | The main focus of this work was to provide a proof of concept, therefore a simple prototype of the retransmitter device was developed just for the relay of information.
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2d9034e112ef1881b6d41bd7b249a7ac59d0c666 | subsection | 14 | 18 | User Interfaces | The solution was tested using two different user interfaces. Instead of using one for the driver and the other for a pedestrian we used both to emulate the behaviour of the driver signalling device. This decision was made, to compare the performance of the smartphone with a dedicate BLE module. Thus, the first user int... | {
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750ef7d3cce2a0c869fb237ce2c6ef2a131cac2b | subsection | 15 | 18 | Experimental Evaluation | In order to validate the operation of the proposed system, two sets of experimental tests were conducted in the field. Firstly, the communications link performance between the centralized controller and the BLE retransmitter was evaluated. For each test run, both devices were statically positioned in specified road loc... | {
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ffff8e672a6ffee8dc2c8f1ba1ade9bfcc34fbb3 | subsection | 16 | 18 | Experimental Evaluation | However, in order to cope with the real-time demands of a safety-critical road traffic system, the performance of the personal VTL system can be greatly improved by employing dedicated hardware, such as the utilized BLE module plus the Raspberry Pi. | {
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39a7cd4b49289d793c4036ce2a4bd0feceb46236 | subsection | 17 | 18 | Conclusions | Going beyond the existing state-of-the-art, in this paper, we proposed a new architecture for virtual traffic light systems. The proposed architecture has unique features since it exploits the current ubiquity of smart devices, deploying a virtual traffic light system as close to the user as possible. Hence, users can ... | {
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12c48031da1ff83778a513443468fb6e5932001a | abstract | 0 | 24 | Abstract | The Emu Chick is a prototype system designed around the concept of migratory
memory-side processing. Rather than transferring large amounts of data across
power-hungry, high-latency interconnects, the Emu Chick moves lightweight
thread contexts to near-memory cores before the beginning of each memory read.
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dae333a2c68a21c4ecf017ebaa0be9d080c8cef8 | subsection | 1 | 24 | Introduction | Analysis of data represented as graphs, sparse tensors, and other non-regular structures poses many challenges for traditional computer architectures because the data locality of these applications typically occurs in small bursts. While individual data elements may have multiple associated attributes nearby (e.g. neig... | {
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65e491bd1eb3526285356728b390c38c9efe25c6 | subsection | 2 | 24 | The Emu Architecture | The Emu architecture focuses on improved random-access bandwidth scalability by migrating lightweight, Gossamer threads or “threadlets” to data and emphasizing fine-grained memory access.
A general Emu system consists of the following processing elements, as illustrated in Figure REF :A common stationary processor runs... | {
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6e3e5b08a50be5e1198ecf0b8d511cfb6d3e7ef9 | subsection | 3 | 24 | Emu Chick Prototype | The Emu Chick prototype is still in active development. The current hardware iteration uses an Arria 10 FPGA on each node card to implement the Gossamer cores, the migration engine, and the stationary cores. Several aspects of the system are scaled down in the current prototype with respect to the next-generation syste... | {
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0f862f39aa4db93e5f13687eb37eed9a584b3552 | subsection | 4 | 24 | Emu Simulator | Emu provides a timing simulator implemented using SystemC, and this simulator can be used to test and evaluate software before running on the hardware. Previous characterization experiments in employed a configuration of the simulator to match the characteristics of a single node (8 nodelets) of the Chick hardware for ... | {
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56f4ba7a10a180916517ba15f26b02a07399c05f | subsection | 5 | 24 | Common CPU-Focused Comparison Platform | In order to make an initial comparison of the Emu's memory bandwidth characteristics with commodity hardware, each benchmark is also run on an four-socket Intel Xeon E7-4850 v3 (Haswell) machine with 2 TiB of DDR4 (referred to as Haswell Xeon in associated results). The CPUs on the Haswell server are each clocked at 2.... | {
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c9bbb2d233b10f8fbef277738a076ce73148d9c3 | subsection | 6 | 24 | Metrics for Comparing the Emu Prototype with Cache-Based Hardware | The architectural design choices that enable the Emu computational model (migrate threads instead of data, narrow memory channels, limited thread context) and the base platforms for the prototype (FPGAs with lower clock frequencies) make it difficult to accurately
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9f4f86d5fb934d9adaef1c41bbee9b7911e4bddd | subsection | 7 | 24 | Benchmarks | As discussed in Section REF , the Emu Chick toolchain currently lacks support for cilk_for and Cilk reducers. However, we present several benchmarks that use Cilk semantics to characterize the performance of the system, specifically focusing on kernels that expose the memory bandwidth characteristics of the system and ... | {
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d0af17fd6f2587b465086f8c866346a2135d5624 | subsection | 8 | 24 | STREAM | The STREAM benchmark has been ported and tuned for the Emu hardware to measure raw memory bandwidth. The ADD kernel computes the vector sum of two large arrays of 8-byte integers, storing the result in a third array. On the Emu Chick these arrays are striped across all the nodelets in the system.This benchmark demonst... | {
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93184f58590371fd85842dded60ce408ba02606d | subsection | 9 | 24 | STREAM | The Emu bandwidth is currently limited by CPU speed and thread count rather than DDR bus transfer rates. However even with this prototype system we can observe improvements in other benchmarks where the memory access pattern is not as linear and predictable as it is with STREAM. | {
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1b22762a58f05baee0127e222e70be19e7905152 | subsection | 10 | 24 | Pointer Chasing | In this benchmark, each thread adds up all the elements in a linked list. Each element consists of an 8-byte payload and an 8-byte pointer to the next element. After the elements of this linked list are grouped into blocks, their ordering is randomized. This permutation may be applied to the ordering of the elements wi... | {
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d198975ce5ff4c201bc3296945b87d996f290b98 | subsection | 11 | 24 | Pointer Chasing | The performance of each system has been normalized to the peak measured bandwidth of the system (i.e., the best result on the STREAM benchmark). In the pointer chasing benchmark, the Emu system is much better at using the available system bandwidth, using 65% of available system bandwidth in most cases and 25% in the ... | {
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849c424dfcc89195ecef5ab8c7382138efdfe129 | subsection | 12 | 24 | Sparse Matrix-Vector Multiplication (SpMV) | In addition to being a fundamental kernel for graph analytics and sparse tensor decomposition applications, SpMV provides an opportunity to investigate data layout strategies on the Emu's global physical address space. Emu provides a “local” malloc (mw_localmalloc) similar to a traditional contiguous malloc as well as ... | {
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b056efb05cb7cec54418e230453853febaf76275 | subsection | 13 | 24 | Results | The updated characterization of the Emu Chick repeats the STREAM, pointer chasing, and SpMV experiments that were initially investigated in , but these experiments focus on further characterizing the entire Chick “multi-node” system that uses all 64 nodelets (across 8 nodes) within the Chick. Single node results are pr... | {
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0014c275b1454ad5fab57b27b669e775fc7e5081 | subsection | 14 | 24 | Sparse Matrix-Vector Multiplication | We use a synthetic Laplacian matrix as an input corresponding to a d-dimensional k-point stencil on a grid of length n in each dimension. d=2 and k=5 specify a sparse matrix generated from a 5-point, 2-D, n \times n stencil. With the Laplacian size n, the matrix is n^2 \times n^2 with 5 diagonals.
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d8f277b2ccd5acf26f7170355b47be8021a05b48 | subsection | 15 | 24 | Sparse Matrix-Vector Multiplication | Furthermore, x86 NUMA interleaved layouts have much larger granularity for striping (pages versus elements in an array on the Emu), which likely also penalizes the “emu (interleaved)" implementation.Following the Haswell Xeon results, we compare the total percentage of STREAM bandwidth that is achieved for SpMV on the ... | {
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0afc79e1ed9b767dbe281ef8f40a3ad9f4582717 | subsection | 16 | 24 | Discussion | This characterization raises important topics for programming memory-centric architectures like the Emu Chick and also for building realistic comparisons between prototype novel architectures and existing architectures. | {
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e7cae0e333d5267542f259d6dfe4da3701b54621 | subsection | 17 | 24 | Achievable Bandwidth for the Current Emu Chick Prototype | Using STREAM, pointer chasing, and SpMV, we provide an initial look at the performance of the Emu Chick for these fundamental operations. However, these results also point to a fundamental issue with the initial Emu Chick prototype. The design of the Chick with 1 Gossamer Core per nodelet and 8 nodelets per node leads ... | {
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f4e9e07448dce046186d8333762e21ab59f06229 | subsection | 18 | 24 | Achievable Bandwidth for the Current Emu Chick Prototype | Moreover, looking at the Chick's memory system design, we see that the measured results for STREAM ADD are 8x slower than the NCDIMM's theoretical peak achievable bandwidth.This analysis points to one conclusion - the current Emu Chick prototype is compute-bound for all microbenchmarks due to a low number of GCs and by... | {
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7c08d1c394b6e31efaf81d2c7f39502dd098717f | subsection | 19 | 24 | Caveats for Programming the Emu Chick | While Cilk provides an easy entry point for programming microbenchmarks for the Chick, our initial characterization has demonstrated some pitfalls for obtaining good performance on the Chick prototype. Primarily, the programmer must consciously design algorithms that optimize data layouts across nodes and that limit lo... | {
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ade130134712181cfe050809fd0b048e3135e8fd | subsection | 20 | 24 | Caveats for Programming the Emu Chick | While we do not yet have enough application experience with the Emu Chick to fully define new metrics, we propose that there may be promise in focusing on comparison metrics that highlight the differences listed above. For example, a cache-less system like the Emu Chick may not actually move data physically across the ... | {
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64c80ac222360a6c2a0e96e59b15ed00c0b70ec0 | subsection | 21 | 24 | Caveats for Programming the Emu Chick | Other recent work has also looked to extend from low-level characterizations like those presented here by providing initial Emu-focused implementations of Breadth-First Search,
Jaccard index computation
, bitonic sort, and compiler optimizations like loop fusion, edge flipping, and remote updates to reduce migrations
.... | {
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6f33d5c330243cf9c5c48efe4df97209fe76cda6 | subsection | 22 | 24 | Related Work | Advances in memory and integration technologies provide opportunities for profitably moving computation closer to data.
Some proposed architectures return to the older processor-in-memory (PIM) and “intelligent RAM” ideas.
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a7ab50bce61d576bcb5f1494c72d145bbf87db59 | subsection | 23 | 24 | Conclusion | Our microbenchmark evaluation of the Emu Chick demonstrates some of the limitations of the existing prototype system as well as some potential
benefits for massive data analytics applications like streaming graph analytics and sparse tensor decomposition.
We demonstrate multi-nodelet (64 nodelets across 8 nodes) perfor... | {
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fc0b1c81eddbf2909a8897542bb85f7a3e5e1c26 | abstract | 0 | 35 | Abstract | The web ecosystem is rapidly evolving with changing business and functional
models. Cloud platforms are available in a SaaS, PaaS and IaaS model designed
around commoditized Linux based servers. 10 billion users will be online and
accessing the web and its various content. The industry has seen a convergence
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71f4529f4acc757f14c9a866f4d90c497ce69fb0 | subsection | 1 | 35 | Introduction | Web N.
Open source has enabled the development of more efficient internet systems. As application performance is a top constraint, profiling is used to verify the performance of a multi-process and multi-threaded workloads. Additionally, a good developer always makes the optimal use of the platform architecture resourc... | {
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33e6054a9c7485c4f31e005904c3e45a5e3e5f09 | subsection | 2 | 35 | Queries | We evaluate the customer application in the internet. We developed a full-system simulator to evaluate web workloads in a server client environment including network. We found opportunity for the use of virtualization technology to efficiently utilize cloud resources. Additionally, use the virtualization layers for per... | {
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b43aca3d1343f89f0e5bd4cdc853ca4c89869f05 | subsection | 3 | 35 | Assumptions related to functionality / requirements | There is a host of cloud computing infrastructure deployed on Linux based platforms. Linux is open source and supports benchmarking and profiling of various applications. Additionally, it supports the use of Virtualization like Xen and VMware. We use OOP languages including Java, C++ and python. A NoSQL database MongoD... | {
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95ef6140f1e76c0dbff593a5745d245884da3ff7 | subsection | 4 | 35 | Application workloads today | In the broader context of the internet it is always beneficial to host resources close to the client consumption including providing a larger bandwidth to the consumer. Additionally, open platforms and standards enable for a balanced distribution of available bandwidth resources allowing for a scalable platform for 10 ... | {
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7d429c94545af040314f25cc9ab6d7f8acdaea5e | subsection | 5 | 35 | Modeling Notation | Unified Modeling Language (UML)
Class diagram
Use case
Sequence
Eclipse
JDeveloper
JavaScript waviSee Figure REF .
[Figure: Tomcat MongoDB server.] | {
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9867b2eb6f74e9c59122f8ec2e32553ad6b393ac | subsection | 6 | 35 | Alternative options considered | There are a wide range of profiling tools for the Linux kernel.Data collection
sysstat package – iostat, pidstat
sar, atop
Online data - top
iotop, iftop
Tracing – strace, perf_events, mutrace, ftrace
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... | |
c84edc64a71d4882f5a7780c91afdf50c52ac703 | subsection | 8 | 35 | Rationale behind suggested solution | Perf supports performance modeling of a variety of events in the user application and kernel code. We are able to capture this data in the OS and output it to the customer in a web interface. The other set of utilities do not provide sufficient information profiling other than compute utilization and availability of me... | {
"cite_spans": []
} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
] | [
"cs.SE",
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] | 2,018 | en | Computer Science | [
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3ca8011bc1ebcbcafdb6606098d8f2aef2b413cf | subsection | 9 | 35 | Estimation | Java KLOC per week = 0.85
Python KLOC per week = 0.75
[Figure: NO_CAPTION][Table: WBS] | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
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5aac1273add1bec973ab56ca82de33c8d5248245 | subsection | 10 | 35 | SDLC model to be used | We use an Agile, continuous development SDLC model. Incremental development is used to deliver the product in short iterations of 1 to 4 weeks. Incremental delivery includes functions and features that have been developed. Continuous integration is used to integrate work frequently. Each integration is verified and tes... | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
] | [
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e88ab96b832a3bd4463831ded5a36c608f9edd6d | subsection | 11 | 35 | Rationale | Agile development and continuous integration. Enabled early detection and integration of defect and ensures code quality. Development happens in short iterations with fully automated regression tests. High level functional requirements are documented as user stories. Software development follows a model of Figure REF ... | {
"cite_spans": []
} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
] | [
"cs.SE",
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ea90f02dc94668647a3c6713c0d4eb6b8069c5fd | subsection | 12 | 35 | Suggested Customizations proposed | We would use Lean techniques and Kanban. These ensure standardization and balance of the development environment. | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
] | [
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c6aedbdee61f83a9da032d08061ad5428da70beb | subsection | 13 | 35 | Architectural (non-functional) requirements | Interoperability
Scalability
Quality of Service
Time, Cost and Productivity
Distributed Complex Sourcing
Faster Delivery of Innovation
Increasing Complexity
SaaS vs. PaaS
Number of users, usage model
Reuse of Web Services
Agile, continuous development
QOS is supported in tiered cloud service provider | {
"cite_spans": []
} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
] | [
"cs.SE",
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] | 2,018 | en | Computer Science | [
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6333d27ca6af8589db77253af59f805802bfc765 | subsection | 14 | 35 | Critical success factors | Compatibility of development environments.
Availability of source code and High-level architecture specification.
Service Oriented Architecture (SOA)
Linux based application hosted to run in a cloud platform.Platforms while constantly updating themselves there is a standardization of functionality around the Linux o... | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
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33978e74c2e7ca5e4d3798dbe58e2818044597d6 | subsection | 15 | 35 | Architectural overview of the requirement and the feasibility in Linux operating system | Linux supports a host of virtualization and performance profiling tools . A virtualized architecture consists of – Figure REF .Data collection
sysstat package – iostat, pidstat
sar, atop
Online data - top
iotop, iftop
Tracing – strace, perf_events, mutrace
Application profiling. perf. gprof
Virtualization per... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/1952682.1952686",
"end": 73,
"openalex_id": "https://openalex.org/W2103848646",
"raw": "J. Du, N. Sehrawat, and W. Zwaenepoel. Performance profiling of virtual machines. Acm Sigplan Notices, 46(7):3–14, 2011.",
"source_ref_id"... | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
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2f9d74c8383e18ea00fb0b17eed06f9b4e4448ef | subsection | 16 | 35 | Linux Perf | Perf is a performance profiling tool for Linux. It supports trace functionalities for various file system, block layer and syscall events. Tracepoints and instrumentation points are placed at logical locations in the application and kernel code. These have negligible overhead and are used by the perf command to collect... | {
"cite_spans": []
} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
"Shaun C. D'Souza"
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"cs.SE",
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0bac42c0060bb6a290dd0fe28ddf9de52092f4b6 | subsection | 17 | 35 | Intel VTune | VTune is a proprietary performance profiling utility . It supports the implementation of performance counters that are used to profile the application. It supports a GUI and command line interface. It is capable of monitoring thread level and process level performance. It supports compute performance, threading, scalab... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 54,
"openalex_id": "",
"raw": "R. K. Malladi. Using intel® vtune™ performance analyzer events/ratios & optimizing applications. http:/software. intel. com, 2009.",
"source_ref_id": "ec4c76f7e1e72527cc7cab7dc1fc7ea208d16611",... | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
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