In this paper, we consider a symmetric pure jump Markov process X on a
general metric measure space that satisfies volume doubling conditions. We
study estimates of the transition density p(t,x,y) of X and their
stabilities when the jumping kernel for X has general mixed polynomial
growths. Unlike [24], in our setting, the rate function which gives growth of
jumps of X may not be comparable to the scale function which provides the
borderline for p(t,x,y) to have either near-diagonal estimates or
off-diagonal estimates. Under the assumption that the lower scaling index of
scale function is strictly bigger than 1, we establish stabilities of heat
kernel estimates. If underlying metric measure space admits a conservative
diffusion process which has a transition density satisfying a general
sub-Gaussian bounds, we obtain heat kernel estimates which generalize [2,
Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the
rate function and the function F related to walk dimension of underlying
space. As an application, we proved that the finite moment condition in terms
of F on such symmetric Markov process is equivalent to a generalized version
of Khintchine-type law of iterated logarithm at the infinity