Institute for Systems Research
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Item Large Deviations for Partial Sum Processes Over Finite Intervals(1997) Banege, Lionel; Makowski, Armand M.; ISRWith 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.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 .
] 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 $(D[0,1]^p, au_0)$.
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.
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Item Large Deviations for Partial Sum Processes on Infinite Time Intervals with Applications to Single-Server Queues and Effective Bandwidths(1996) Banege, Lionel; Makowski, Armand M.; ISR; CSHCNIn this dissertation, we establish large deviations results for partial sum processes on infinite time intervals, and apply them to the characterization of the large deviations behavior of the stationary and transient output processes of a single-server queue with time-varying capacities. We first show that the extension of a partial sum process on the infinite time interval [0, infinity) satisfies the Large Deviations Principle (LDP) in the function space D [0, infinity), provided the partial sum process itself satisfies the LDP in the space D [0,1]. Furthermore, for a stationary random sequence whose associated partial sum process satisfies the LDP in D [0,1], we establish LDP jointly for a partial sum process based on the entire past and future of sequence, a result especially useful in queueing theory. Through a functional approach at the sample path level, the Contraction Principle then enables us to derive the sample path LDP for processes of interest in the study of single-server queues, from that of the inputs. Finally, using our results, we refine the newly introduced notion of effective bandwidths.