Under minimal moment conditions complete asymptotic expansions are obtained for expectations of unbounded functions of a finite family of independent random variables in the Poissonian setting when the distributions of the random variables have large atoms at zero.

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We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.Keywords: limit theorems, stochastic integral, multiple stochastic integral, elementary stochastic measure, Gaussian processes, stationary sequences of random variables, mixing, U -and V -statistics Statement of the Main ResultsIn the paper we study the limit behavior of the canonical von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The approach uses the representation of such statistics as multiple stochastic integrals with respect to the corresponding normalized empirical product-measure as well as the results of [1]. These results allow us to interpret the corresponding limit random element as a multiple stochastic integral of the kernel of the statistic under consideration with respect to a Gaussian noise which plays a role of the weak limit for the above-mentioned normalized empirical measures.Let {X i ; i ∈ Z} be a stationary sequence of random variables with values in an arbitrary measurable space {X, A } and marginal distribution P . Consider a measurable function f (t 1 , . . . , t d ) : X d → R. Define the von Mises statistic (or V -statistic) by the formula V n := n −d/2 1≤i 1 ,...,i d ≤n f (X i 1 , . . . , X i d ), n = 1, 2, . . . ,where d ≥ 2 and the subscripts i k range independently over the integers from 1 to n, and the function f satisfies the degeneracy condition Ef (t 1 , . . . , t k−1 , X k , t k+1 , . . . , t d ) = 0 (2) for all t 1 , . . . , t d ∈ X and k = 1, . . . , d. The function f is called a kernel of a von Mises statistic, and the statistics with degenerate kernel are called canonical.In the case of independent observations {X i } such statistics were studied in the second half of the last century (see the references and examples of such statistics in [2]). Some limit theorems for these statistics were obtained for the first time by von Mises [3] and Hoeffding [4]. Moreover, in these articles there were introduced the so-called U -statistics: U n := n −d/2or U 0 n := n −d/21≤i 1 <···