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I wonder who first suggested and defined weak stationarity, and which paper it is. It seems that many paper using it as given, but I'm just curious of how it is defined and proved, and the discussions over the definition.

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It was developed by Khintchine in Korrelationstheorie der stationare stochastischen Processe, Math. Ann. 109, 604-615.

As $\rm [I]$ notes:

The second line of development began with a series of papers in 1932-1934 by the Russian mathematician Khintchine who introduced both stationary and weakly stationary stochastic processes and developed the correlation theory for weakly stationary processes [see Khintchine (1934)]. This development was important not only for time series analysis but was also one of the pioneering works in the modern theory of stochastic processes. Later, Kolmogorov (1941a) developed the geometric theory of weakly stationary time series and Cramér (1942) discovered the important spectral decomposition of weakly stationary processes…


Reference:

$\rm [I]$ The Spectral Analysis of Time Series, Lambert H. Koopmans, Academic Press, $1995, $ sec. $2.1, $ p. $29.$

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  • $\begingroup$ It seems unlikely that this is the original, because it gives short shrift to George Udny Yule, who studied these processes a generation or two before 1932. $\endgroup$
    – whuber
    Mar 13, 2023 at 18:51
  • $\begingroup$ Thanks @whuber for pointing this out. I would then look for the specific paper/reference of Yule which contains the development of the relevant concepts. $\endgroup$ Mar 14, 2023 at 0:23

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