Danijel Krizmanić

Weak convergence of partial maxima processes in the M1 topology

It is known that for a sequence of independent and identically distributed random variables (Xn) the regular variation condition is equivalent to weak convergence of partial maxima Mn=max{X1,…,Xn}

Functional weak convergence of partial maxima processes

M_{n}= \max \{X_{1}, \ldots , X_{n}\}

Weak convergence of multivariate partial maxima processes

, appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space of càdlàg functions endowed with the Skorohod J1 topology. We first show that weak convergence of partial maxima Mn holds also for a class of weakly dependent sequences under the joint regular variation condition. Then using this result we obtain a corresponding functional version for the processes of partial maxima Mn(t)=∨i=1⌊nt⌋Xi,t∈[0,1]

A functional limit theorem for dependent sequences with infinite variance stable limits

M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]

Functional limit theorems for weakly dependent regularly varying time series

, but with respect to the Skorohod M1 topology, which is weaker than the more usual J1 topology. We also show that the M1 convergence generally can not be replaced by the J1 convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given.

A Multivariate Functional Limit Theorem in Weak M_{1} Topology

For a strictly stationary sequence of nonnegative regularly varying random variables (Xn) we study functional weak convergence of partial maxima processes Mn(t)=∨i=1⌊nt⌋Xi,t∈[0,1]

A limit theorem for moving averages in the α-stable domain of attraction

M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]

A note on joint functional convergence of partial sum and maxima for linear processes

in the space D[0, 1] with the Skorohod J1 topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J1 and M1 functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition A(an)

On functional weak convergence for partial sum processes

\mathcal {A}(a_{n})

Functional convergence for moving averages with heavy tails and random coefficients.

with the time component.