  Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that
the jump distribution density has moderate tails, i.e., several density moments
are finite, including the second one. The global (for all $x$ and $t$)
asymptotic behavior at infinity of the transition probability (fundamental
solution of the corresponding parabolic convolution operator) is found. Front
propagation of ecological waves in the corresponding population dynamics models
is described.
