The conditions $x>EX/\sqrt{EX^2}$ and $EX\ge0$ imply $x>0$. So, $t[bX_i -x(X_i^2+b^2)/2]$ is a quadratic polynomial in $X_i$ whose leading coefficient $-tx/2$ is less than $0$ (except for the trivial case $t=0$). Therefore this polynomial is bounded from above, and hence the random variable $e^{t[bX_i -x(X_i^2+b^2)/2]}$ is bounded. So, its expectation is finite.
Iosif Pinelis
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