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This module introduces stationarity, such as strict sense stationarity (SSS) and wide sense stationarity (WSS).

Stationarity in a Random Process implies that its statistical characteristics do not change with time . Put another way, if one were to observe a stationary random process at some time t it would be impossible to distinguish the statistical characteristics at that time from those at some other time t .

Strict sense stationarity (sss)

Choose a Random Vector of length N from a Random Process:

X X t 1 X t 2 X t N
Its N th order cdf is
F X ( t 1 ) ,     X ( t N ) x 1 x N X t 1 x 1 X t N x N
X t is defined to be Strict Sense Stationary iff:
F X ( t 1 ) ,     X ( t N ) x 1 x N F X ( t 1 + c ) ,     X ( t N + c ) x 1 x N
for all time shifts c , all finite N and all sets of time points t 1 t N .

Wide sense (weak) stationarity (wss)

If we are only interested in the properties of moments up to 2nd order (mean, autocorrelation, covariance, ...), which isthe case for many practical applications, a weaker form of stationarity can be useful:

X t is defined to be Wide Sense Stationary (or Weakly Stationary) iff:

  • The mean value is independent of t , for all t
    X t μ
  • Autocorrelation depends only upon τ t 2 t 1 , for all t 1
    X t 1 X t 2 X t 1 X t 1 τ r X X τ
Note that, since 2nd-order moments are defined in terms of2nd-order probability distributions, strict sense stationary processes are always wide-sense stationary, but not necessarily vice versa .

Questions & Answers

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Akash Reply
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s. Reply
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are you nano engineer ?
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Source:  OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3
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