On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals $$\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}$$ { X k ( u ) } k ∈ Z , $$u \in \mathbf{I}^d$$ u ∈ I d , $$d \ge 1$$ d ≥ 1 , the normalized partial sums $$\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )$$ S ˇ n t ( u ) = n - 1 / 2 ( X 1 ( u ) + ⋯ + X ⌊ n t ⌋ ( u ) ) , $$t \in [0,1]$$ t ∈ [ 0 , 1 ] and its polygonal version $${S}_{nt}(u)$$ S n t ( u ) are considered under a weak dependence assumption and $$p > 2$$ p > 2 moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process $$\check{S}_{nt}(\widehat{\theta })$$ S ˇ n t ( θ ^ ) , where $$\widehat{\theta } \xrightarrow {\mathbb {P}} \theta $$ θ ^ → P θ , and weaker moment conditions ( $$p = 2$$ p = 2 if $$d = 1$$ d = 1 ) are assumed.