We collect together a number of examples of random walk
where the characteristic function of the first step has a
singularity at the point t=0. The function \log\varphi(t) has
two different expansions for positive and negative $t$ near the
origin; we call the coefficients of these expansions left and
right quasicumulants. Such examples include the trace of a
two dimensional random walk {(X_n,Y_n)} on the x-axis, and the
subordinated random walk (X_{\tau_n}) where (\tau_n) is an
appropriate sequence of random times. Using quasicumulants we derive an asymptotic expansion for the distribution of the sums of i.i.d. random variables, and assuming
further differentiability condition we are able to give sharp
estimate in the variable x of the remainder term.