On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals $$\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}$${Xk(u)}k∈Z, $$u \in \mathbf{I}^d$$u∈Id, $$d \ge 1$$d≥1, the normalized partial sums $$\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )$$Sˇnt(u)=n-1/2(X1(u)+⋯+X⌊nt⌋(u)), $$t \in [0,1]$$t∈[0,1] and its polygonal version $${S}_{nt}(u)$$Snt(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 $$\check{S}_{nt}(\widehat{\theta })$$Sˇnt(θ^), where $$\widehat{\theta } \xrightarrow {\mathbb {P}} \theta $$θ^→Pθ, and weaker moment conditions ($$p = 2$$p=2 if $$d = 1$$d=1) are assumed.

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