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"Theorem A" in this paper by Y. Gordon: http://www.math.uiuc.edu/~mjunge/Comps/Gordonm.pdf is a comparison inequality for Gaussian processes:

Is there an analogue of this result for subgaussian processes? By "analogous" I mean to have the process $X_{ij}$ subgaussian with bounded subgaussian norm, and the same conclusion up to constant factors.

That is, something that would recover Talagrand's comparison inequality for $n=1$ (Corollary 8.5.3 in http://www-personal.umich.edu/~romanv/teaching/2015-16/626/HDP-book.pdf).

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  • $\begingroup$ Analogous in what sense? Its assumption or conclusion? $\endgroup$
    – Henry.L
    Commented Apr 4, 2017 at 20:37
  • $\begingroup$ I edited the question to explain. $\endgroup$
    – axk
    Commented Apr 4, 2017 at 21:53
  • $\begingroup$ Okay, I misunderstood what you mean by analogous, in that direction you need to consult some treatise on Sobolev/Orlicz spaces. I guess. $\endgroup$
    – Henry.L
    Commented Apr 4, 2017 at 21:59

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I think a closest inequality in controlling the deviance between two processes is McDiarmid’s Inequality, and it has an extension which also applies to subgaussian random variables Concentration in unbounded metric spaces and algorithmic stability.

You need to describe what kind of analogy you are looking for in order for me to add more.

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