A PROBABILISTIC MODEL FOR CASH FLOW

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1 A PROBABILISTIC MODEL FOR CASH FLOW MIHAI N. PASCU Communicated by the former editorial board We introduce and study a probabilistic model for the cash ow in a society in which the individuals decide to keep the money or spend it based on coin ips i.i.d., Bernoulli random variables. Alternately, the model is suitable for an Economy in which the rms decide to pay or not the other rms according to the same rule. We show that in this model the trajectory of a coin a monetary unit is a recurrent irreducible martingale, and we determine its corresponding scaling limit. More precisely, we derive the Strong Law of Large Numbers, the Central Limit Theorem and the Functional Central Limit Theorem for the corresponding random walk described by the trajectory of the coin. We also discuss the economic implications of the model and some possible further extensions of the model. AMS 2010 Subject Classication: 60G50, 60F05, 60F15, 60F17, 60J70. Key words: random walk, martingale, Strong Law of Large Numbers, Central Limit Theorem, Functional Central Limit Theorem, probabilistic model. 1. INTRODUCTION Several authors studied the eects of various moral beliefs the so-called golden rulessee for example [Be1], [Be2], [Be3] and the references cited therein in interactive games among neighbors. In the present paper, we derive a model for the cash ow i.e., the path described by a monetary unit, for example a penny in a society in which the individuals appeal to one of the two rules: pass it along to thy neighbor the givers or keep it for thyself the keepers. We further assume that the decisions of the individuals are independent of each other, occur with the same probability, and are also independent on previous decisions. Alternately, the model is also suitable for an Economy in which the companies adopt, at each instant of time, one of the two possible strategies: to pay or to keep the money. In the present paper, we consider that a monetary unit for example a penny is initially located at the origin, and we study the random walk corresponding to the trajectory of the coin within the population a random walk in a random environment. MATH. REPORTS 1565, , 97106

2 98 Mihai N. Pascu 2 The structure of the paper is the following. In Section 2 we introduce the probabilistic model for the cash ow, and we set up the notation. In the next section, in Lemma 3.1 we consider the process V n representing the number of transitions of the penny between dierent states up to time n N, and we show that V n increases a.s. to innity. Using this, and the classical Strong Law of Large Numbers, in Proposition 3.2 we show that the random walk S n representing the position of the penny at time n N is a recurrent martingale. Theorem 3.3 contains three limit theorems for random walk S n : the Strong Law of Large Numbers SLLN, the Central Limit Theorem CLT, and the Functional Central Limit Theorem FCLT. The paper concludes with a discussion of the practical implications of the model, and with some possible extensions of it. 2. THE MODEL We consider a population in which the individuals occupy the integer position on the real line. We consider that when given a monetary unit a penny, the individuals of the population decide to keep it or to spend it, more precisely to give it to on of their adjacent neighbors. Assuming that the times when the transitions occur are discrete, we construct and study the model based on the following assumptions. The individuals keep or pass away the coin with the same probability also constant in time, the decision being independent on previous decisions, and also independent on the decisions of the rest of the population. If an individual decides to pass away the coin, he gives it to one of his neighbors, with equal probability. Mathematically, under the hypothesis above, the random walk S n representing the position of the coin within the population at time n N can be described as follows. On a xed probability space Ω, F, P, consider the following sequences of i.i.d. random variables, also independent of each other: i Y i i N, taking the values ±1 with equal probability Y i represents the increment of the position of the penny at time i, if the individual possessing it is willing to pass it one of his neighbors; ii U i,j i Z,j N -Bernoulli random variables taking the value 1 with probability p 0, 1 U i,j = 1 if the individual i posses the penny at time j and he is willing to pass it to one of his neighbors, and U i,j = 0 otherwise. Considering that the coin is initially located at the origin, the position of the penny in this model is given by the random walk S n n N, where S 0 = 0,

3 3 A probabilistic model for cash ow 99 S n = X X n, n 1, and 2.1 X n+1 = U Sn,nY n+1 = { Yn+1 if U Sn,n = 1 0 otherwise, n N. We also consider the ltration F =F n n N, where F 0 = σu i,0 : i Z and F n = σu i,j, Y k : i Z, j < n, k n, n 1 represent the σ-algebra of events known up to time n N. Note that in this model we consider that the decision of an individual to pass the coin to his left/right neighbor is taken at ctitious time n + 1/2, in other words it is known at the discrete time n + 1 but not known at time n. Also note that according to this denition, U i,n 1 and Y n are F n -measurable random variables, and that U i,n and Y n+1 are independent of F n, for all i Z and n N. Finally, we also consider the process V n dened by n V n = 1 {Sj S j 1 } = U Sj,j, n 1, j=1 representing the number of transition of the random walk between distinct sites, up to time n. j=0 3. MAIN RESULTS With this preamble we can now prove the rst result, as follows. Lemma 3.1. Almost surely we have lim n V n =. Proof. Since U i,j i,j N is an i.i.d. sequence of Bernoulli random variables, we have P U Sj,j = 1 = i Z P U i,j = 1, S j = i = i Z P U i,j = 1 P S j = i = p i Z P S j = i = p, for all j N, by the independence of U i,j and S j S j is F j measurable, and U i,j is independent of F j. Since p > 0, we obtain P U Sj,j = 1 =, j=0 so the conclusion of the lemma follows by the second Borel-Cantelli lemma, provided we show that A j = { U Sj,j = 1 }, j N, forms a sequence of independent events.

4 100 Mihai N. Pascu 4 However, this follows easily using the independence of the sequence U i,j, as follows. For any 0 i < j we have P U Si,i = 1, U Sj,j = 1 = k,z P U Si,i = 1, U k,j = 1, S j = k = k Z P U Si,i = 1, S j = k P U k,j = 1, since U Si,i and S j are F j -measurable random variables recall that U k,i is F i+1 - measurable random variable and by hypothesis i + 1 j, and U k,j is independent of F j. We obtain P U Si,i = 1, U Sj,j = 1 = k Z P U Si,i = 1, S j = k P U k,j = 1 = p k Z P U Si,i = 1, S j = k = pp U Si,i = 1 = P U Si,i = 1 P U Sj,j = 1, which shows that the events A j j N are pairwise independent. Using a similar argument and mathematical induction it can be shown that the events A j j N are also independent, concluding the proof. The above lemma shows that outside a set of probability zero we can dene the right inverse α n of V n by 3.1 α n = min {m 0 : V m n}, n N. Recall see for example [No] that a Markov chain is called irreducible if starting at any point in the state space it visits any other point in the state space with positive probability, and is called recurrent if it returns a.s. to its starting point for any choice of the starting point in the state space. Some of the properties of the random walk S n are contained in the following. Proposition 3.2. The random walk S n n N is a recurrent irreducible F n n N -martingale. and Proof. First note that by the denitions of S n and α n, we have S αn = = S αn+1 1 S αn+1 P S αn+1 S αn = ±1 = 1 2, for all n N, so the time-changed random walk S αn n N is a symmetric simple random walk on Z.

5 5 A probabilistic model for cash ow 101 By a theorem due to Pólya, the symmetric simple random walk on Z is recurrent see for example [Bi], pp This, together with the fact that by Lemma 3.1 the process V n is a.s. increasing to innity hence the same holds true for its inverse α n, shows that the random walk S n n N is also recurrent. To prove the second claim, since S n is recurrent, it suces to show that starting at the origin S n will hit any integer k Z with positive probability. But this follows easily using the following lower bound P n 1 : S n = k P S i = i sgnk, i = 1,..., k = P U i 1,i 1 = 1, Y i = sgnk, i = 1,..., k p k = > 0, 2 for any k Z. To prove the last claim, consider the σ-algebra F n = σ U i,j, Y j : i Z, j n = σ F n {U i,n : i N} F n generated by F n and the random variables U i,n i Z, and note that S n is F n - measurable, U Sn,n is F n -measurable, and Y n+1 is independent of F n. Using the properties of conditional expectation, we obtain concluding the proof. E S n+1 F n = S n + E X n+1 F n = S n + E E U Sn,nY n+1 F n Fn = S n + E U Sn,nE Y n+1 F n Fn = S n + E U Sn,nE Y n+1 F n = S n + E Y n+1 E U Sn,n F n = S n + 0 E U Sn,n F n = S n, The main results for the random walk S n introduced in Section 2 are contained in the following. Theorem 3.3. The following hold true for the random walk S n n N. SLLN Almost surely we have S n 3.2 lim n n = 0. CLT We also have 3.3 S n np D Z stably,

6 102 Mihai N. Pascu 6 where Z is a standard normal random variable. FCLT Moreover, if ξ n t 0 t 1 is the continuous process ξ n t = 1 np S k + X k+1 nt k, k n t k + 1 n composed of the straight line segments joining the points k = 0, 1,..., n 1, k n, S k np 0 k n, then all nite dimensional distributions of ξ n t converge weakly as n to those of a standard Brownian motion B t 0 t 1 starting at the origin. Proof. By the proof of the previous proposition, S αn n N is a symmetric simple random walk. By the classical SLLN see for example [Bi, Theorem 6.1] we have a.s. S αn 3.4 lim n n = 0. Since by Lemma 3.1 the process V n increases a.s. to, the above a.s. convergence also holds along the subsequence of indices V n. That is, a.s. we also have S αvn lim = 0. n V n Note that in the above it is possible that V n = 0 for some indices n = 1, 2,..., so the sequence Sα Vn V n is not dened for these indices n. However, by Lemma 3.1 V n is a.s. increasing to innity, so the rst few terms of the sequence Sα Vn V n are not well dened only on a set of zero probability, which we ignore alternately, we can dene Sα Vn V n = 0 if V n = 0. By the denition of the nondecreasing process V n and its inverse α n it is easy to see that if α Vn = m for some m = m ω N, then V m = = V n, so S m = = S n ; this shows that S αvn = S m = S n, and combining with the above we obtain S n 3.5 lim = 0 a.s. n V n In the proof of Lemma 3.1 we shown that the events {U Si,i = 1} i N are independent. A similar proof shows that the events {U Si,i = a i } i N are also independent for any choice of a i {0, 1}, i N, and therefore U Si,i i N is an independent sequence of random variables. Since U Si,i i N are also identically distributed with mean E U S0,0 = p, using again the classical SLLN it follows that a.s. we have V n 3.6 lim n n = lim n n 1 U Si,i i=0 n = EU S0,0 = p.

7 7 A probabilistic model for cash ow 103 Using 3.5 and 3.6 we obtain S n lim n n = lim S n V n lim n V n n n = 0 p = 0 with probability 1, which concludes the rst part of the proof. Next, note that since by Proposition 3.2 S n is a F n -martingale, and using the classical terminology and notation see for example [Ha, Chapter 3], it follows that {S ni, F ni, 1 i k n, n 1} is a martingale array, where k n = n, s n = Var S n, F ni = F i and S ni = s 1 n S n. We have n 1 s 2 n = Var S n = E X 2 i = E U 2 S i 1,i 1Y 2 i = E i=0 U Si,i = np, so the corresponding martingale increments are given by X ni = S ni S n,i 1 = s 1 n X i = 1 np X i, 1 i k n, n 1. In order to prove the last claim of the theorem, we will apply the Martingale Central Limit Theorem MCLT to the martingale array S ni. For an arbitrary ε > 0 we have P max X ni > ε = P max U Si 1,i 1Y i > ε np = 1 i k n 1 i k n = P max U Si 1,i 1 > ε np 1 i k n for any n 1 pε 2, which shows that P 3.7 max 1 i k n X ni P 0. By the rst part of the proof we have a.s. therefore we obtain lim n k n Xni 2 1 = lim n np a.s., and in particular 3.8 E U 2 S i 1,i 1Y 2 i k n Xni 2 Finally, for all n 1 we also have max Xni 2 = 1 1 i k n np E P 1. max US 2 1 i k i 1,i 1Yi 2 n max U Si 1,i 1 > 1 1 i k n 1 lim n n n 1 i=0 = 0 U Si,i = p, and n 1 1 = lim U Si 1,i 1 = 1 n np = 1 np E i=0 max 1 i k n U Si 1,i 1 1 np,

8 104 Mihai N. Pascu 8 which shows that 3.9 E max Xni 2 1 i k n is bounded in n. The relations above show that the hypotheses of the MCLT are met see for example [Ha, Theorem 3.2], and therefore we obtain S nkn = S n np Z stably, where the random variable Z has a standard normal N 0, 1 distribution, concluding the proof of the second claim. To prove the last claim we will use the Martingale Functional Central Limit Theorem MFCLT. By Proposition 3.2, S n is a F n -martingale with S 0 = 0, and for any n 1 we have σn 2 = E Xn 2 F n 1 = E USn 1,n 1 F n 1 = E U k,n 1 1 {Sn 1 =k} Fn 1 k Z = k Z 1 {Sn 1 =k}e U k,n 1 F n 1 = k Z 1 {Sn 1 =k}e U k,n It follows that n = p k Z 1 {Sn 1 =k} = p. σi 2 = np and s2 n = E σi 2 = np, so trivially σi 2 E σi 2 Also note that 1 s 2 E X 2 1 i 1 { Xi εs n} = n np = 1 np = 1 P 1. E US 2 i 1,i 1Yi 2 1 { U Si 1,i 1 ε } np E U Si 1,i 11 { U Si 1,i 1 ε } = 0 np

9 9 A probabilistic model for cash ow 105 for all n > 1 ε 2 p since U i,j {0, 1} for all i Z, j N. In particular, it follows that the Lindeberg condition holds for S n, that is we have s 2 n E Xi 2 P 1 { Xi εs n} 0. The last claim of the theorem follows now from MFCLT see for example [Br, Theorem 2] using , concluding the proof of the theorem. 4. CONCLUDING REMARKS Aside from their importance in their own right, the results obtained in the previous section have also practical Economic importance. For example, the fact that by Proposition 3.2 the random walk is recurrent shows that an individual possessing the coin at some time, is sure to receive again the coin innitely often in the future. From the Economic point of view this is reassuring, for it shows a good circulation of money within the society. The fact that the random walk described by the coin is irreducible shows that the model is fair, in the sense that all the individuals of the society will be in the possession of the coin at some time. Also, the fact that the random walk is a martingale shows that the model is fair there is no particular tendency for the coin to favor a certain region of the society. Finally, the convergence results in Theorem 3.3 can be used for practical purposes, to compute various probability related to the random walk described by the coin. The main idea is here that under the appropriate rescaling 1 time by a factor of n, space 1 by np, and for large values of n, the probabilities concerning the random walk S n can be approximated by the corresponding probabilities of a standard 1-dimensional Brownian motion. We conclude with a possible extension of the model. In the constructing the model in Section 2 we assumed that the decisions of the individuals to keep/give away the coin were modeled by Bernoulli random variables with the same parameter p 0, 1. A possible extension of the model is to consider the case when for each individual the decisions are still modeled by Bernoulli random variables, but not necessary with the same parameter for all individuals. A rst problem here is that the corresponding random walk will not be in general irreducible/recurrent, at least without additional hypotheses on the corresponding probabilities. This extension of the model would be perhaps closer to the real-world situation, since in general individuals in a society have dierent tendencies probabilities to keep/give away money.

10 106 Mihai N. Pascu 10 Acknowledgements. The author kindly acknowledges the support from the Sectorial Operational Programme Human Resources Development SOP HRD, nanced from the European Social Fund and by the Romanian Government under the contract SOP HRD/89/1.5/S/ REFERENCES [Be1] T.C. Berstrom, Ethics, evolution, and games among family and neighbors. Manuscript online at [Be2] T.C. Bergstrom, Some Evolutionary Economics of Family Partnerships. Am. Econ. Rev , 2, [Be3] T.C. Bergstrom. On the evolution of altruistic ethical rules for siblings. Am. Econ. Rev , 1, [Bi] P. Billingsley, Probability and Measure. John Wiley & Sons, Inc., New York, [Br] B.M. Brown, Martingale central limit theorems. Ann. Math. Statist , 1, [Ha] P. Hall and C.C. Heyde, Martingale Limit Theory and its Application. Academic Press, Inc., New YorkLondon, [No] J.R. Norris, Markov Chains. Cambridge University Press, Cambridge, Received 9 August 2012 Transilvania University of Bra³ov Faculty of Mathematics and Computer Science Str. Iuliu Maniu Nr. 50, Bra³ov, Romania mihai.pascu@unitbv.ro and Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764, Bucharest, Romania