# Stochastic-Process Limits: An Introduction to by Ward Whitt

By Ward Whitt

This booklet is ready stochastic-process limits - limits during which a series of stochastic strategies converges to a different stochastic strategy. those are necessary and fascinating simply because they generate easy approximations for sophisticated stochastic strategies and likewise aid clarify the statistical regularity linked to a macroscopic view of uncertainty. This ebook emphasizes the continuous-mapping method of receive new stochastic-process limits from formerly tested stochastic-process limits. The continuous-mapping method is utilized to acquire heavy-traffic-stochastic-process limits for queueing types, together with the case within which there are unequalled jumps within the restrict method. those heavy-traffic limits generate easy approximations for sophisticated queueing approaches and so they exhibit the impression of variability upon queueing functionality.

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2) at an arbitrary time t in the interval [0, 1]. More generally, we can 22 1. Experiencing Statistical Regularity √ consider an arbitrary t ≥ 0. To do so, we set cn = n and m = 1/2. 10) for each t ≥ 0, where m = 1/2 and σ 2 = 1/12. 2) must be asymptotically equivalent to c n for some constant c as n → ∞. 4. Indeed, if we instead scale by cn = np for p > 1/2, then the values converge to 0 as n → ∞. Similarly, if we scale by cn = np for p < 1/2, then the values diverge as n → ∞. ) This property can be conﬁrmed by further analysis of simulations, but we do not pursue it.

We want to strengthen the form of convergence in order to be able to deduce convergence of related quantities of interest; in particular, we want to show that plots of the centered random walk converge to plots of standard Brownian motion as n → ∞. ’s). , the k-dimensional marginal distributions for all k. 12) can be extended to obtain (Sn (t1 ), . . , Sn (tk )) ⇒ (σB(t1 ), . . 16) as n → ∞ for all positive integers k and all k time points t1 , . . 9). 16) is a much stronger conclusion. However, we want to go even further.

The empirical cdf of uniform random numbers. To illustrate, we now consider the diﬀerence between the empirical cdf associated with n uniform random numbers on the interval [0, 1] and the uniform cdf itself. Since the uniform cdf is F (t) = t, 0 ≤ t ≤ 1, we now want to plot Fn (t) − t versus t for 0 ≤ t ≤ 1. Since the function Fn (t) − t, 0 ≤ t ≤ 1, is a function of a continuous variable, the plotting is less routine than for the random walk. However, the empirical cdf Fn has special structure, making it possible to do the plotting quite easily.