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I found the explanation for the $\lesssim$ in this question:

What does a tilde underneath an inequality mean?

My problem is of the following nature:

$g_t(y)$ is a stochastic function and $M_t$ some stochastic process, where $t\in[0,T]$ is the time parameter. Then there is the growth condition \begin{align*} |g_t(y)|\lesssim 1+|y|+|M_t|. \end{align*} Now, when squaring $g_t(y)$ I wonder, whether the growth condition expands to \begin{align*} |g_t(y)|^2\lesssim 1+|y|^2+|M_t|^2 \text{ or }\\ |g_t(y)|^2 \lesssim (1+|y|+|M_t|)^2. \end{align*}

The reason for my question is, that I found an inequation, where they used the first inequation. This triggered the thought, that maybe $g_t(y)$'s growth is determined in the different parameters independently and thus one can square them individually.

Leoncino
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1 Answers1

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Both options are valid, because for any non-negative real numbers $a,b,c$, we have $$a^2+b^2+c^2\le(a+b+c)^2\le 3(a^2+b^2+c^2)$$ So each of $1+|y|^2+|M_t|^2$ and $(1+|y|+|M_t|)^2$ is bounded by a constant multiple of the other.

TonyK
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