We consider a sequence of i.i.d. random variables,

(ξ) = {(ξ_{i})}_{i = 0, 1, 2, \dots}

E ξ_{0} = 0

E ξ_{0}^{2} = 1

, and subordinate it by a doubly stochastic Poisson process

Π (λ t)

, where

λ \geq 0

is a random variable and

Π

is a standard Poisson process. The subordinated continuous time process

ψ (t) = ξ_{Π (λ t)}

is known as the PSI-process. Elements of the triplet

(Π, λ, (ξ))

are supposed to be independent. For sums of n, independent copies of such processes, normalized by

\sqrt{n}

, we establish a functional limit theorem in the Skorokhod space

D [0, T]

, for any

T > 0

, under the assumption

E | ξ_{0} |^{2 h} < \infty

for some

h > 1 / γ^{2}

. Here,

γ \in (0, 1]

reflects the tail behavior of the distribution of

λ

, in particular,

γ \equiv 1

when

E λ < \infty

. The limit process is a stationary Gaussian process with the covariance function

E e^{- λ u}

u \geq 0

. As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales.

10,960 Results Found

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