Strict stationarity of a process defined as the product of lags of another process

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Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting issue.



Assume $X_t$ is a discrete time stochastic process. Define the stochastic process $Y_t$ as $Y_t:=X_tX_t-1$.



1. Question. If $Y_t$ is strictly stationary, can we conclude that $X_t$ must be strictly stationary as well?



2. Follow-on question. How would the answer change if instead $Y_t:=X_tX_t-1X_t-2cdots$?




stochastic-processes stationary-processes




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  • Let $X_t$ alternate between zero and one deterministically.
    – spaceisdarkgreen
    Jul 20 at 19:30










  • Omg. Thanks a lot @spaceisdarkgreen.
    – Martinigale
    Jul 21 at 10:07














up vote
1
down vote

favorite












Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting issue.



Assume $X_t$ is a discrete time stochastic process. Define the stochastic process $Y_t$ as $Y_t:=X_tX_t-1$.



1. Question. If $Y_t$ is strictly stationary, can we conclude that $X_t$ must be strictly stationary as well?



2. Follow-on question. How would the answer change if instead $Y_t:=X_tX_t-1X_t-2cdots$?




stochastic-processes stationary-processes




share|cite|improve this question



















  • Let $X_t$ alternate between zero and one deterministically.
    – spaceisdarkgreen
    Jul 20 at 19:30










  • Omg. Thanks a lot @spaceisdarkgreen.
    – Martinigale
    Jul 21 at 10:07












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting issue.



Assume $X_t$ is a discrete time stochastic process. Define the stochastic process $Y_t$ as $Y_t:=X_tX_t-1$.



1. Question. If $Y_t$ is strictly stationary, can we conclude that $X_t$ must be strictly stationary as well?



2. Follow-on question. How would the answer change if instead $Y_t:=X_tX_t-1X_t-2cdots$?




stochastic-processes stationary-processes




share|cite|improve this question











Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting issue.



Assume $X_t$ is a discrete time stochastic process. Define the stochastic process $Y_t$ as $Y_t:=X_tX_t-1$.



1. Question. If $Y_t$ is strictly stationary, can we conclude that $X_t$ must be strictly stationary as well?



2. Follow-on question. How would the answer change if instead $Y_t:=X_tX_t-1X_t-2cdots$?





stochastic-processes stationary-processes





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 20 at 18:34









Martinigale

63




63











  • Let $X_t$ alternate between zero and one deterministically.
    – spaceisdarkgreen
    Jul 20 at 19:30










  • Omg. Thanks a lot @spaceisdarkgreen.
    – Martinigale
    Jul 21 at 10:07
















  • Let $X_t$ alternate between zero and one deterministically.
    – spaceisdarkgreen
    Jul 20 at 19:30










  • Omg. Thanks a lot @spaceisdarkgreen.
    – Martinigale
    Jul 21 at 10:07















Let $X_t$ alternate between zero and one deterministically.
– spaceisdarkgreen
Jul 20 at 19:30




Let $X_t$ alternate between zero and one deterministically.
– spaceisdarkgreen
Jul 20 at 19:30












Omg. Thanks a lot @spaceisdarkgreen.
– Martinigale
Jul 21 at 10:07




Omg. Thanks a lot @spaceisdarkgreen.
– Martinigale
Jul 21 at 10:07















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