Most likely paths to error when estimating the mean of a reflected random walk

It is known that simulation of the mean position of a Reflected Random Walk (RRW) $\{W_n\}$ exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let $I$ denote the rate function for the one dimensional increment process. If $I$ is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, $\psi^*$, of the paths $n^{-1} W_{\lfloor tn\rfloor}$ is to be zero apart from on an interval $[T_0,T_1]\subset[0,1]$ and to satisfy the functional equation \begin{align*} \nabla I\left(\ddt\psi^*(t)\right)=\lambda^*(T_1-t) \quad \text{whenever } \psi(t)\neq 0. \end{align*} If $I$ is non-coercive, a similar, but slightly more involved, result holds. These results prove, in broad generality, that Monte Carlo estimates of the steady-state mean position of a RRW have a high likelihood of over-estimation. This has serious implications for the performance evaluation of queueing systems by simulation techniques where steady state expected queue-length and waiting time are key performance metrics. The results show that na\"ive estimates of these quantities from simulation are highly likely to be conservative.Comment: 23 pages, 8 figure

On the tail asymptotics of the area swept under the Brownian storage graph

In this paper, the area swept under the workload graph is analyzed: with $\{Q(t) : t\ge0\}$ denoting the stationary workload process, the asymptotic behavior of $$\pi_{T(u)}(u):={\mathbb{P}}\biggl(\int_0^ {T(u)}Q(r)\,\mathrm{d}r>u\biggr)$$ is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of $\pi_{T(u)}(u)$ are given for the case that $T(u)$ grows slower than $\sqrt{u}$, and then logarithmic asymptotics for (i) $T(u)=T\sqrt{u}$ (relying on sample-path large deviations), and (ii) $\sqrt{u}=\mathrm{o}(T(u))$ but $T(u)=\mathrm{o}(u)$. Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ491 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Most likely paths to error when estimating the mean of a reflected random walk

It is known that simulation of the mean position of a Reflected Random Walk (RRW) {Wn} exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let I denote the rate function for the one dimensional increment process. If I is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, ∗, of the paths n−1W⌊tn⌋ is to be zero apart from on an interval [T0, T1] ⊂ [0, 1] and to satisfy the functional equation ∇

Most likely paths to error when estimating the mean of a reflected random walk

It is known that simulation of the mean position of a reflected random walk {Wn} exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let I denote the rate function for the one dimensional increment process. If I is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, ψ, of the paths n −1 W ⌊tn ⌋ is to be zero apart from on an interval [T0, T1] ⊂ [0, 1] and to satisfy the functional equation ∇I ( d dt ψ(t)) = λ ∗ (T1 − t) whenever ψ(t) ̸ = 0. If I is non-coercive, a similar, but slightly more involved, result holds

Tail asymptotics for busy periods

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta}, \quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. II) The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by, [\psi^*(t) = -\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have very different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.Comment: 15 pages, 5 figure

Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a ∣x∣|x| potential

We study the fluctuations of the area $A=\int_0^T x(t) dt$ under a one-dimensional Brownian motion $x(t)$ in a trapping potential $\sim |x|$, at long times $T\to\infty$. We find that typical fluctuations of $A$ follow a Gaussian distribution with a variance that grows linearly in time (at large $T$), as do all higher cumulants of the distribution. However, large deviations of $A$ are not described by the ``usual'' scaling (i.e., the large deviations principle), and are instead described by two different anomalous scaling behaviors: Moderatly-large deviations of $A$, obey the anomalous scaling $P\left(A;T\right)\sim e^{-T^{1/3}f\left(A/T^{2/3}\right)}$ while very large deviations behave as $P\left(A;T\right)\sim e^{-T\Psi\left(A/T^{2}\right)}$. We find the associated rate functions $f$ and $\Psi$ exactly. Each of the two functions contains a singularity, which we interpret as dynamical phase transitions of the first and third order, respectively. We uncover the origin of these striking behaviors by characterizing the most likely scenario(s) for the system to reach a given atypical value of $A$. We extend our analysis by studying the absolute area $B=\int_0^T|x(t)| dt$ and also by generalizing to higher spatial dimension, focusing on the particular case of three dimensions.Comment: 18 pages, 4 figure

Explicit Mean-Square Error Bounds for Monte-Carlo and Linear Stochastic Approximation

This paper concerns error bounds for recursive equations subject to Markovian disturbances. Motivating examples abound within the fields of Markov chain Monte Carlo (MCMC) and Reinforcement Learning (RL), and many of these algorithms can be interpreted as special cases of stochastic approximation (SA). It is argued that it is not possible in general to obtain a Hoeffding bound on the error sequence, even when the underlying Markov chain is reversible and geometrically ergodic, such as the M/M/1 queue. This is motivation for the focus on mean square error bounds for parameter estimates. It is shown that mean square error achieves the optimal rate of $O(1/n)$, subject to conditions on the step-size sequence. Moreover, the exact constants in the rate are obtained, which is of great value in algorithm design