Lemma 1 is widely used in the stability proof of stochastic process.
Lemma 1 Assume that $\xi_k$ is a stochastic process and there is a stochastic process $V(\xi_k)$ as well as real numbers $\upsilon_{\min}$,$\upsilon_{\max}$, and $0<\lambda<1$，such that $\forall k$
$$
(a) \quad \upsilon_{\min}\left \|\xi_k \right\|^2 \leq V(\xi_k) \leq \upsilon_{\max}\left \|\xi_k \right\|^2
$$
$$
(b) \quad E[ V(\xi_{k+1})\big |\xi_{k}] - V(\xi_k) \leq \mu-\lambda V(\xi_k)
$$
are fulfilled.Then the stochastic process is bounded in mean square, i.e.,
$$
E[\left \|\xi_k \right\|^2] \leq \frac{\upsilon_{\max}}{\upsilon_{\min}}E[\left \|\xi_0 \right\|^2](1-\lambda)^k + \frac{\mu}{\upsilon_{\min}}\sum_{i=1}^{k-1}(1-\lambda)^{i}
$$
The question is, if the condition b in Lemma 1 is modified as
$$
(b) \quad E[ V(\xi_{k+1})\big |\xi_{k}] - V(\xi_k) \leq \mu-\lambda V(\xi_k)+\gamma \xi_k
$$
Then if the stochastic process is $\xi_k$ still bounded in mean square? And what conditions parameter $\mu$,$\lambda$,$\gamma$ need to satisfy?
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$\begingroup$ if possible, can you include some references for the above objects, lemma and its proof? Then we can try to modify it. $\endgroup$– Thomas KojarCommented Nov 5, 2022 at 18:09
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$\begingroup$ Thank you in advance, the proof of this lemma contains a combination of [1, Sec. 4.1, Th. 1] and [48, Th.2] 1.R. G. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAM J. Contr., vol. 9, pp. 372–384, 1971 2.T. J. Tarn and Y. Rasis, “Observers for nonlinear stochastic systems,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 441–448, 1976 $\endgroup$– themarshalCommented Nov 9, 2022 at 8:26
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$\begingroup$ in the Agniel/Jury work, if you define a new Wk in (28) by now removing gamma xi_k, do you lose supermartingale? $\endgroup$– Thomas KojarCommented Nov 9, 2022 at 17:27
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$\begingroup$ if possible, add the above references in your post and also some specific attempts and where you got stuck. $\endgroup$– Thomas KojarCommented Nov 9, 2022 at 17:29
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