  In this paper, we consider a symmetric pure jump Markov process $X$ on a
metric measure space with volume doubling conditions. Our focus is on
estimating the transition density $p(t,x,y)$ of $X$ and studying its stability
when the jumping kernel exhibits general mixed polynomial growth.
  Unlike previous work, in our setting, the rate function governing the jump
growth may not be comparable to the scale function that determines whether
$p(t,x,y)$ has near-diagonal or off-diagonal estimates. Under the assumption
that lower scaling index of scale function is greater than $1$, we establish
stabilities of heat kernel estimates. Additionally, if the metric measure space
admits a conservative diffusion process with a transition density satisfying
sub-Gaussian bounds, we generalize heat kernel estimates from [3, Theorems 1.2
and 1.4] using the rate function and the function $F$ related to walk dimension
of underlying space. As an application, we prove the equivalence between a
finite moment condition based on $F$ and a generalized Khintchine-type law of
iterated logarithm at infinity for symmetric Markov processes.
