Cluster sets for partial sums and partial sum processes

Abstract: Let

X, X_{1}, X_{2}, l d o t s

be i.i.d. mean zero random vectors with values in a separable Banach space

B

,

S_{n} = X_{1} + c d o t s + X_{n}

for

n g e 1

, and assume

c_{n} : n g e 1

is a suitably regular sequence of constants. Furthermore, let

S_{(n)} (t), 0 l e t l e 1

be the corresponding linearly interpolated partial sum processes. We study the cluster sets

A = C (S_{n} / c_{n})

and

m a t h c a l A = C (S_{(n)} (c d o t) / c_{n})

. In particular,

A

and

m a t h c a l A

are shown to be nonrandom, and we derive criteria when elements in

B

and continuous functions

f : [0, 1] o B

belong to

A

and

m a t h c a l A

, respectively. When

B = m a t h b b R^{d}

we refine our clustering criteria to show both

A

and

m a t h c a l A

are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for

m a t h c a l A

. When the coordinates of

X

in

m a t h b b R^{d}

are independent random variables, we are able to represent

m a t h c a l A

in terms of

A

and the classical Strassen set

m a t h c a l K

, and, except for degenerate cases, show

m a t h c a l A

is strictly larger than the lower bound set whenever

d g e 2

. In addition, we show that for any compact, symmetric, star-like subset

A

of

m a t h b b R^{d}

, there exists an

X

such that the corresponding functional cluster set

m a t h c a l A

is always the lower bound subset. If

d = 2

, then additional refinements identify

m a t h c a l A

as a subset of

(x_{1} g_{1}, x_{2} g_{2}) : (x_{1}, x_{2}) i n A, g_{1}, g_{2} i n m a t h c a l K

, which is the functional cluster set obtained when the coordinates are assumed to be independent.

Cites Work