Large Deviations for Partial Sum Processes Over Finite Intervals

Abstract

With any sequence {xn, n = ﯱ, ﯲ, ...} of IRp -valued random variables, we associate the partial sum processes {XTN(.)}$ which take value, in the space $(D[0,T]^p, au_0)$ of, right- continuous functions $[0,T] ightarrow R^p$ with left-hand, limits equipped with Skorohod's $J_1$ topology.<P>Furthermore, in an attempt to capture the past of the sequence, we introduce the negative partial sum processes ${ X_n^{T,-}(cdot),; n=z }$ defined by [ X_n^{T,-}(t)(omega) ~ equiv ~ left{ a{ll} ds {1 over n} {sum_{i=1-ceiling{nt}}^{0} x_i(omega)} & mbox{if} quad ceiling{nt} geq 1 \ 0 & mbox{otherwise} ea ight., quad t in [0,T], quad omega in Omega .<P>] These processes take value in the space $(D_l[0,T]^p, au_0)$ of, left- continuous functions $[0,T] ightarrow R^p$ with right-hand, limits also equipped with the Skorohod's $J_1$ topology. This paper explores some of the issues associated with, transfering the LDP for the family ${X_n^1 (cdot),~n=z}$ in $(D[0,1]^p, au_0)$ to the families ${X_n^{T}(cdot), ; n=z}$ in $(D[0,T]^p, au_0)$, ${X_n^{T,-}(cdot), ; n=z}$ in $(D_l[0,T]^p, au_0)$ and ${(X_n^T(cdot), X_n^{T,-}(cdot)), ; n=z}$ in $(D[0,T]^p, au_0) x (D_l[0,T]^p, au_0)$ for arbitrary $T>0$; the last two types of transfers require, stationarity of the underlying sequence ${x_n, ; n=pmz}$. The motivation for this work can be found in the study of, large deviations properties for general single server queues, and more specifically, in the derivation of the effective bandwidth, of its output process, all discussed in a companion paper. In a significant departure from the situation under the uniform topology, such transfers are not automatic under the Skorohod topology, as additional continuity properties are required on the elements of, the effective domain of the rate function $I_X$ of the LDP, for ${X_n^1(cdot),~n=z}$ in $(D[0,1]^p, au_0)$.<P>However, when the rate function $I_X$ is of the usual integral form, the transfers are automatic, and the new rate functions assume, very simple forms suggesting that from the perspective, of large deviations, the past of the underlying stationary process, behaves {it as if} it were independent of its future.<P>======================================================= ===========