Linear Recurrences via Probability

The long run behavior of a linear recurrence is investigated using standard results from probability theory. In a recent paper [1] in the American Mathematical Monthly, the authors look at several distinct approaches to studying the convergence and limit L of a sequence (xn) given by a linear recurrence. Our approach is to use probability to study the same sequence. The nth term xn of the sequence is expressed as the average value of a random process observed at time n. The convergence of (xn) then follows from the asymptotic stability of the underlying random process, meaning intuitively that the random process approaches the same equilibrium for any initial value. We give two different probabilistic interpretations of (xn) with the asymptotic stability deriving from, respectively, the ergodic theorem for Markov chains and the renewal theorem for random walks. To be more concrete, fix two sequences of real numbers (αk)k=1 and (ak) m k=1. For 1 ≤ n ≤ m let xn = an , while for n > m let xn = αm xn−1 + αm−1xn−2 + · · · + α1xn−m . (1) This recurrence can be expressed using linear algebra as follows. Define an m × m matrix P = ⎛ ⎜⎜⎜⎝ 0 1 0 · · · 0 0 0 0 1 · · · 0 0 .. .. .. · · · .. .. 0 0 0 · · · 0 1 α1 α2 α3 · · · αm−1 αm ⎞ ⎟⎟⎟⎠ , and two vectors of length m: e1 = (1, 0, . . . , 0) and a = (a1, a2, . . . , am) . Then for every n ≥ 1, we can write xn = e1 Pn−1 a. (2) We introduce probability by assuming that αk ≥ 0 with ∑m k=1 αk = 1. If, in addition, we take αm > 0, then the limit L exists and has a natural interpretation. Indeed, L is necessarily a weighted average of the initial constants (ak)k=1, and the probabilistic viewpoint allows us to understand and interpret those weights. 1. MARKOV CHAIN MODEL. Consider a particle that starts at 1 then moves up through 2, 3, . . . one at a time until it reaches m, at which time it jumps back down to state k, where k ≤ m, and the new state is chosen with probability αk . The particle then continues back up toward m and repeats the process indefinitely. http://dx.doi.org/10.4169/amer.math.monthly.122.04.386 MSC: Primary 40A05, Secondary 60J10; 60G50 386 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 122 More formally, this particle is a Markov chain (Xn) with state space {1, 2, . . . ,m} and transition matrix P . If f : {1, 2, . . . ,m} → R is given by f (k) = ak , then Equation (2) says xn = E( f (Xn)), for n ≥ 1. (3) That is, xn is the average f value associated with the state of the Markov chain observed at time n. Since αm > 0, the ergodic theorem for Markov chains [2, Theorem 8.18] gives xn = E( f (Xn)) → ∫ f (s) π(ds), where π is the limiting probability distribution of the chain on {1, 2, . . . ,m}. In other words, we may write the limit as L = ∑mk=1 ak π(k). The π-value for any state can be understood as the asymptotic percentage of time that the Markov chain spends in that state. Figure 1 shows that the Markov chain spends most of its time in state m because on each return to state m it jumps down to some state k and then loops back to state m. Some of the loops are short, some are long, but every one of them contains a visit to state m. States on the left like 1 or 2 are not included in every loop and so are visited less often than states like m − 1 or m.