I need the result of following problem $$E_z[\exp(-a(\frac{1}{z})^b)]$$ where $a>0,b>0$ and $z$ is defined as below (I do not know if there is other simpler way of defining $z$ but if somebody can work over it then please simplify this definition).
Definition of $z$: Mathematically it is defined as $$z=\sum_{i\in \Phi-{\text{first element in } \Phi}}k_ip_i^{-m}$$ here $p_i$ is distance of the $i$th point in the poisson point process to the origin, $m>0$ and all $k_i$'s are i.i.d exponential random variables with parameter $\lambda$. $\Phi$ represents the Poisson point process. It can be observed from the above equation that summation starts from the second point from the origin. I know how to simplify this equation for $b=-1$ but in this problem $b>0$ hence if somebody know how to solve it or if somebody can provide some references to look then I will be very thankful. Thanks in advance.
Edit: Since exponential is a convex function hence I think we can also get a lower bound on the above expression with the help of following identity $$f(E_z(z))\leq E_z[f(z)]$$ where $z$ in the subscript denotes the random variable while $z$ in the parenthesis represent the actual realization of $z$.
I do not know which background you are from, but I shall try to answer it from the wireless networks point of view.
The term $z$ is exactly the interference experienced by the typical user (a user at the origin), where $k_i$ denotes the Rayleigh fading channel power gain and $m$ is the path loss exponent. It is assumed that the typical user connects to the closest node and all other nodes cause interference to the user. Given this background, there are few excellent references that deal with the moments of interference, which are as follows:
[1] https://rkganti.wordpress.com/2011/10/15/interference-in-random-wireless-networks/
[2] Stochastic Geometry for Wireless Networks by Martin Haenggi
Reference [1] can help you get the moments of the interference.