 dimension $d$ changes, provided that as $d$ changes, we do not introduce larger variances into the underlying multinormal distribution itself.
The sizes of the variances of the underlying distribution are controlled in this simulation by making the largest eigenvalue of the covariance matrix equal to $1$. This keeps the probability density "cloud" within bounds as the dimension increases, no matter what the shape of this cloud might be.  Simulations of other models of behavior of the system as the dimension increases can be created simply by changing how the eigenvalues are generated; one example (using a Gamma distribution) is shown commented out in the R code below.
What we are looking for is to verify that the standard deviations of the parameter estimates do not appreciably change when the dimension $d$ is changed. I therefore show the results for two extremes, $d=2$ and $d=60$, using the same amount of data ($30$) in both cases.  It is noteworthy that the number of parameters estimated when $d=60$, equal to $1890$, far exceeds the number of vectors ($30$) and exceeds even the individual numbers ($30*60=1800$) in the entire dataset.
Let's begin with two dimensions, $d=2$.  There are five parameters: two variances (with standard deviations of $0.097$ and $0.182$ in this simulation), a covariance (SD = $0.126$), and two means (SD = $0.11$ and $0.15$).  With different simulations (obtainable by changing the starting value of the random seed) these will vary a bit, but they will consistently be of comparable size when the sample size is $n=30$.  For instance, in the next simulation the SDs are $0.014$, $0.263$, $0.043$, $0.04$, and $0.18$, respectively: they all changed but are of comparable orders of magnitude.
(These statements can be supported theoretically but the point here is to provide a purely empirical demonstration.)
Now we move to $d=60$, keeping the sample size at $n=30$.  Specifically, this means each sample consists of $30$ vectors, each having $60$ components.  Rather than list all $1890$ standard deviations, let's just look at pictures of them using histograms to depict their ranges.

The scatterplots in the top row compare the actual parameters sigma ($\sigma$) and mu ($\mu
dimension $ d$ changes, provided that as $ d$ changes, we do not introduce large variance into the underlying multinormal distribution itself. 
 The sizes of the variances of the fundamental distribution are controlled in this simulation by have the largest eigenvalue of the covariance matrix equal to $ 1$. This keep open the probability density " cloud " within limit as the dimension increases, no matter what the shape of this swarm might be.   Simulations of other model of behavior of the system as the property increases can be created simply by switch how the eigenvalues are generated; one example (using a Gamma distribution) is show commented out in the R code below. 
 What we are looking for is to verify that the standard deviations of the argument estimates do not appreciably change when the dimension $ d$ is changed. I therefore show the result for two extremes, $ d=2 $ and $ d=60 $, using the same amount of data ($ 30 $) in both cases.   It is noteworthy that the number of parameters estimated when $ d=60 $, equal to $ 1890 $, far exceed the number of vectors ($ 30 $) and surpass even the individual numbers ($ 30 * 60=1800 $) in the entire dataset. 
 Let's begin with two dimension, $ d=2$.   There are five parameters: two variances (with standard deviations of $ 0.097 $ and $ 0.182 $ in this model), a covariance (SD = $ 0.126 $), and two means (SD = $ 0.11 $ and $ 0.15 $).   With different simulations (obtainable by change the starting value of the random seed) these will vary a bit, but they will consistently be of comparable size when the sample size is $ n=30$.   For case, in the next simulation the SDs are $ 0.014 $, $ 0.263 $, $ 0.043 $, $ 0.04 $, and $ 0.18 $, respectively: they all changed but are of comparable orders of order of magnitude. 
 (These statements can be supported theoretically but the point here is to provide a purely empirical demonstration .) 
 Now we move to $ d=60 $, keep the sample distribution size at $ n=30$.   Specifically, this means each sample consists of $ 30 $ vectors, each having $ 60 $ component.   Rather than number all $ 1890 $ standard deviations, let's barely look at pictures of them use histograms to picture their ranges. 

 The scatterplots in the top course compare the actual parameters sigma ($ \sigma$) and mu ($ \mu
 dilension $d$ changes, providtd that as $d$ chanyws, we vo not jntroducd larger variances into the nndeelyint multinormal distribugion itsepf.
The sizes id the variehces of the uhferlviig distribution are contrmlled in this vioupation by making the largest eigenvwlue of tje covariance iatrpx equzl to $1$. This keeps the probability sensity "cloud" within bounds as the dimension ijcrewses, no matter whah the shape of eyis cloud mieht be.  Simulations of other models of behavior of thd syscem as the einendhon increasxs can be created simply bf changong how the einenvanuew are generated; one eeample (using a Gamma distribudikn) is shown commebtwd ouj in dhe F coae gekos.
What ae ere looking for is to cerify that the stamdwgc deviations of thq [arameter estimates do not appreciably bhanfe when the dimension $d$ is changed. I therefote show thq results for two extremes, $d=2$ and $d=60$, using the same amouit of dwgq ($30$) in both cases.  It is noteworthy that the numfsr on parameters estlmated when $d=60$, equak ho $1890$, far exceeds jhe numysr of vectors ($30$) and edceeds gven tye indiviqual numbers ($30*60=1800$) in the entire datqset.
Let's begpn wuth two dimensions, $d=2$.  There axe fivg paraketers: two variances (wich stahdard deviahions of $0.097$ xnd $0.182$ in this simjlanion), a covariance (SD = $0.126$), and two means (SD = $0.11$ aud $0.15$).  Witf digferene simulatilns (ondainable by changijg thg starding value of the random seed) these will tery a bit, but tvey will cousistektly be of compwrable size whgn the saiple rize is $n=30$.  For invtance, in tre next simuldjion the SDs ere $0.014$, $0.263$, $0.043$, $0.04$, agd $0.18$, eespwctivelh: they all chanbed but age of xomparable orders pf ozgnitude.
(These sccttmwnts can be sulpofteq nheprqdically but dhe ooivy herd is to proyidd a lurely empirical demmnstdation.)
Now we move yo $d=60$, keepibg the swmple size at $n=30$.  Specifically, tjis mxans eech sakplg consists of $30$ vectors, each havjng $60$ complnekts.  Rather tran oist all $1890$ stcndard deviations, let's just look at pictnres of them using hisjograms to depict thenr ranges.

The xcattqrplots it the top row comparw the actual paraketers sigma ($\sigma$) ans mu ($\mg
dimension $d$ changes, provided that as $d$ do introduce larger into the underlying of variances of the distribution are controlled this simulation by making the largest of the covariance matrix equal to $1$. This keeps the probability density "cloud" bounds as the dimension increases, no matter what the shape of this cloud be. of models behavior of the system as the dimension increases can be created simply by changing how the are generated; one example (using a Gamma distribution) shown commented out in R code below. What we looking is to that standard of the parameter do not appreciably change when the dimension $d$ is changed. I therefore show the results for two $d=2$ and the same of ($30$) both cases. It that the number of parameters estimated to $1890$, far exceeds the number of vectors and exceeds the individual numbers ($30*60=1800$) in the dataset. Let's begin with two dimensions, $d=2$. There five parameters: two variances (with standard deviations of $0.097$ and $0.182$ in this simulation), a = $0.126$), and two (SD = $0.11$ $0.15$). different (obtainable changing the value of the random seed) these will vary a bit, but will consistently be of comparable size when the sample size For in the next the SDs are $0.014$, $0.043$, and $0.18$, respectively: they but of magnitude. statements be supported theoretically but point here is to provide purely empirical demonstration.) Now the sample size at $n=30$. Specifically, this means sample consists of $30$ vectors, each having components. Rather than list all $1890$ standard deviations, let's just look at of them to depict their ranges. The scatterplots in the row compare the actual sigma ($\sigma$) and mu ($\mu
 dimension $d$ changes, provided That as $d$ chaNges, wE do Not InTrodUce lArger variances INto tHe underlying multinormaL distRiBUtioN ItSelf.
THe sizes OF tHE VarIaNcEs oF tHE uNderlYinG distriBution are cOntRoLled in this siMUlAtion by makIng The largest eiGenValue oF tHe cOVariaNce MatriX equal TO $1$. This kEeps the prObABility DEnsity "cLOUd" WithIn bounds as the dimeNSiON increases, no maTter whAt THe SHApe Of tHis cloud miGhT be.  SiMUlationS Of OTHEr mODels of behavioR of the systeM As tHe dimeNsIon INcreasEs can Be CReaTed simply by ChanGing how thE eigenVAlues arE GeneratEd; one eXamPle (UsinG A GAmMa dIsTRibUTiOn) iS ShoWn commenTeD oUt in tHe R cODE BElow.
whaT we aRe looKing for is to veRifY thaT The StandArd deViatIoNs of tHe paraMeter EsTimates do not appReciAbly changE whEn The DiMensiON $d$ is chAngEd. I TherefoRe show tHE reSuLTS FoR two extremes, $d=2$ and $d=60$, uSiNG ThE same amoUnt of dATa ($30$) In BOth cases.  it Is nOtewORThy thAt thE NuMber of paRameteRS eStImated wHeN $d=60$, equaL tO $1890$, faR exCeeds THe nuMber of Vectors ($30$) aNd excEEds even the indiVIdual numbers ($30*60=1800$) iN ThE ENtIRe daTasEt.
Let's begin With TWo diMensIOnS, $d=2$.  THEre arE five PaRAmETers: two variances (witH sTandarD deviAtions of $0.097$ and $0.182$ in This simulaTION), a covariAnce (sd = $0.126$), aND two means (SD = $0.11$ and $0.15$).  with dIfferent siMUlations (ObtaiNable by cHanging thE STarting vAluE of The RanDOM sEed) these will vARY a biT, bUt they wIll ConsistEntLy bE of ComPaRable size When the sAmPlE sIzE is $N=30$.  For iNStance, in ThE neXt SimUlatiON the SDS are $0.014$, $0.263$, $0.043$, $0.04$, aNd $0.18$, reSpEcTIveLy: they aLL cHANged BuT aRe of ComPaRable OrdeRS of MagnituDe.
(These stAteMEnts CaN bE supporTed theoreticaLlY but the poiNt HerE is to pROVide a purEly empirical demonstratiON.)
Now we mOve To $d=60$, keEpinG the samplE siZe at $n=30$.  SPecIFicallY, this mEans eAcH saMPLe conSIStS of $30$ VeCtors, each hAVIng $60$ CompoNeNts.  RAther thAn list all $1890$ standard dEViaTions, let's just LooK at pICTuRes OF tHEm uSiNG hiSTOgrams to depict tHeir ranges.

thE ScAtterplots IN thE tOp row coMpare thE actuAL parameTers sigma ($\Sigma$) and mU ($\mU
 dimension $d$ changes, pr ovided tha t as$d$  ch an ges,  wedo not introdu c e la rger variances into th e und er l ying mu ltino rmal di s tr i b uti on  i tse lf . 
T he si zes  of the  variances  of  t he underlyin g  d istributio n a re controlle d i n this  s imu l ation  by  maki ng the larges t eigenva lu e  of th e  covari a n ce  mat rix equal to $1$. Th i s keeps the pr obabil it y  d e n sit y " cloud" wit hi n bou n ds as t h ed i m ens i on increases,  no matterw hat  the s ha peo f this  clou dm igh t be.  Simu lati ons of ot her mo d els ofb ehavior  of th e s yst em a s  t he  di me n sio n  i ncr e ase s can be  c re atedsimp l y b y ch ang inghow t he eigenvalue s a re g e ner ated;  oneexam pl e (us ing aGamma  d istribution) is  sho wn commen ted  o utin  theR  codebel ow. 
What w e are l o oki ng f o ris to verify thatth e st andard d eviati o ns  o f  the par am ete r es t i mates  don ot  appreci ably c h an ge  when t he  dimen si on$d$  is c h ange d. I t herefore  show the results fo r  two extremes ,  $ d = 2$ and$d= 60$, usingthes ameamou n tofd ata ( $30$)  i n  b o th cases.  It is no te worthy  that  the number o f paramete r s estimate d wh e n$ d=60$, equal t o $18 90$, far e x ceeds th e num ber of v ectors ($ 3 0 $) and e xce eds  ev ent h eindividual nu m b ers($ 30*60=1 800 $) in t heent ire  da ta set.
Let' s beginwi th  t wo  di mensi o ns, $d=2 $.   T he reare f i ve par amete rs:tw ov ari ances ( w it h stan da rd  dev iat io ns of  $0. 0 97$  and $0 .182$ inthi s  sim ul at ion), a  covariance ( SD  = $0.126$ ),  an d twom e ans (SD= $0.11$ and $0.15$). W ith dif fer ent s imul ations (o bta inable  by changi ng the  star ti ngv a lue o f th e r an dom seed)t h ese  will  v arya bit,but they will cons i ste ntly be of co mpa rabl e si zew he n  th es amp l e  size is $n=30$ .  For ins ta n ce , in the n e xtsi mulatio n the S Ds ar e  $0.014 $, $0.263 $, $0.043 $,  $0. 0 4 $,and $0.18$ , respec tively: t h ey al l  c hange d b ut are  o f c ompar able o r der s ofmagnit ud e.
(Th ese s ta tementscan be supported theore ticall y but  th e point h ere isto provid e apurely emp iri cal  demo nst r ation .)
N o wwem ove t o $d = 60$, keep i ng  th e sa mple size a t $ n=3 0$. Spe c ifical ly,this means each s a mple consistsof $ 3 0 $ v ect o rs,ea ch having $60$  co mp o n ents.  R at her than li st all $ 18 9 0$ st andard  devia tions,l e t' s  justlook  at  pictures  of  t h em usin ghi s togram s to  d epicttheirr ange s . 

The scatterplo ts in t he to p  ro w com pa re thea ctua l paramete rs sigma ($ \sigma $) a nd mu  ($\mu
 dimension_$d$ changes,_provided that as $d$_changes, we_do_not introduce_larger_variances into the_underlying multinormal distribution_itself.
The sizes of the_variances of the_underlying_distribution are controlled in this simulation by making the largest eigenvalue of the covariance_matrix_equal to_$1$._This_keeps the probability density "cloud"_within bounds as the dimension_increases, no_matter what the shape of this cloud might_be._ Simulations of_other models of behavior of the system as the_dimension increases can be created simply_by changing how_the_eigenvalues_are generated; one example_(using a Gamma distribution) is shown_commented out in the R code_below.
What we are looking for is to_verify that the standard deviations of_the parameter estimates do not_appreciably change_when the dimension $d$ is_changed. I therefore_show the_results for two_extremes, $d=2$ and $d=60$, using the_same amount of_data ($30$) in both cases. _It_is noteworthy that_the_number_of parameters_estimated when $d=60$,_equal_to $1890$,_far_exceeds the number of vectors ($30$)_and_exceeds even the individual numbers ($30*60=1800$) in_the entire dataset.
Let's begin_with_two dimensions, $d=2$. _There are five parameters: two_variances (with standard deviations of $0.097$_and $0.182$_in this_simulation), a covariance (SD = $0.126$), and two means (SD =_$0.11$ and $0.15$).  With different_simulations (obtainable by changing_the starting_value_of the random_seed)_these will_vary a bit, but they will consistently_be of_comparable size when the sample size_is $n=30$.  For_instance,_in the next simulation the SDs_are $0.014$, $0.263$, $0.043$, $0.04$, and_$0.18$, respectively: they all changed_but_are_of comparable orders of magnitude.
(These_statements can be supported theoretically but_the point here_is to provide a purely empirical demonstration.)
Now_we_move to $d=60$, keeping the sample_size_at $n=30$.  Specifically, this means_each_sample_consists of $30$ vectors, each_having $60$ components.  Rather than_list all $1890$ standard deviations, let's just look at_pictures of them_using histograms to depict their_ranges.

The_scatterplots_in the top row compare the actual parameters sigma ($\sigma$)_and mu_($\mu