Mixture distributions moments if one distribution has undefined/infinite moments Consider probability density functions $f_{1}\left(x\right)$ and $f_{2}\left(x\right)$ and the mixture distribution
$$f_{3}\left(x\right)\equiv pf_{1}\left(x\right)+\left(1-p\right)f_{2}\left(x\right)$$
Assume that the moments associated with $f_{1}\left(x\right)$ are all clearly defined, but the moments of $f_{2}\left(x\right)$ may not necessarily be (e.g. a Cauchy or Levy distribution). 
To calculate the expected value one would find
$$\int_{-\infty}^{\infty}xf_{3}\left(x\right)dx=p\int_{-\infty}^{\infty}xf_{1}\left(x\right)dx+\left(1-p\right)\int_{-\infty}^{\infty}xf_{2}\left(x\right)dx$$
Presumably if $\int_{-\infty}^{\infty}xf_{2}\left(x\right)dx$ is infinite or undefined, then the expected value only exists when $p=1$.
Is this logic correct, even if $p$ is close to but less than $1$? Does it extend to higher moments as well?
 A: Yes, you are correct. If $X_1 \sim f_1$, $X_2 \sim f_2$, and $X_3 \sim f_3$, then your equation shows that $$E(X_3) = p \cdot E(X_1) + (1-p) \cdot E(X_2)$$ Therefore if either one of $E(X_1)$ or $E(X_2)$ in non-finite/non-existent then $E(X_3)$ will be non-finite/non-existent also if $p \in (0,1)$ - this is true even if $p$ is very near $0$ or $1$. 
To get some intuition for this, note that the mixture distribution can be thought of a drawing from $f_1$ with probability $p$ and $f_2$ with probability $1-p$. Bearing that in mind, take an example where $f_1$ is the density of the reciprocal of a standard normal (a distribution with infinite mean), $f_2$ is the standard normal density and $p$ is some very small value (say $.01$). Consider sampling variables that have density $f_3$ - of the $1\%$ that are drawn from $f_1$, there will be some extreme values characteristic of a distribution with non-finite mean.
You're also correct that this same logic would apply to higher moments - replace $x$ with $x^k$ in your integrals and you can make an exactly analogous argument. Where it gets more complicated is when, for example, both integrals are non-finite but this seems beyond the scope of the question :)
