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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.