A PROBABILISTIC MODEL FOR CASH FLOW MIHAI N. PASCU Communicated by the former editorial board

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 15(65), 1 (2013), 97106

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 Vn 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 Sn

representing the position of the penny at time n∈Nis a recurrent martingale.

Theorem 3.3 contains three limit theorems for random walk Sn: 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 walkS_n repre- senting 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 ±1with equal probability (Y_i represents the increment of the position of the penny at timei, if the individual possessing it is willing to pass it one of his neighbors);

ii)(Ui,j)i∈Z,j∈N-Bernoulli random variables taking the value1with proba- bility p∈(0,1)(U_i,j = 1 if the individualiposses the penny at timej and he is willing to pass it to one of his neighbors, and Ui,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 S₀ = 0,

Sn=X1+· · ·+Xn,n≥1, and (2.1) X_n+1=U_Sn,nY_n+1 =

Y_n+1 if U_Sn,n = 1

0 otherwise , n∈N.

We also consider the ltration F =(F_n)n∈N, where F₀ =σ(Ui,0 :i∈Z) and F_n =σ(Ui,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, Ui,n−1 and Y_n are F_n-measurable random variables, and thatUi,nand Yn+1 are independent ofF_n, for alli∈Z and n∈N.

Finally, we also consider the processV_n dened by

(2.2) Vn=

n

X

j=1

1_{Sj6=Sj−1}=

n−1

X

j=0

U_Sj,j, n≥1,

representing the number of transition of the random walk between distinct sites, up to time n.

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∈Nis an i.i.d. sequence of Bernoulli random variables, we have

P U_Sj,j= 1

=X

i∈Z

P(U_i,j = 1, S_j =i) =X

i∈Z

P(U_i,j = 1)P(S_j =i)

=pX

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 Ui,j is independent of F_j).

Sincep >0,we obtain

∞

X

j=0

P(U_Sj,j = 1) =∞,

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 indepen- dent events.

However, this follows easily using the independence of the sequenceUi,j, as follows. For any 0≤i < j we have

P U_Si,i= 1, U_Sj,j= 1

=X

k,Z

P(U_Si,i= 1, U_k,j = 1, S_j =k)

=X

k∈Z

P(U_Si,i= 1, Sj =k)P(U_k,j= 1), sinceUSi,iandSj areF_j-measurable random variables (recall thatUk,iisF_i+1- measurable random variable and by hypothesisi+ 1≤j), andU_k,j is indepen- dent of F_j. We obtain

P U_Si,i= 1, U_Sj,j = 1

=X

k∈Z

P(U_Si,i= 1, S_j =k)P(U_k,j = 1)

=pX

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(Aj)_j∈

Nare pairwise independent. Using a similar argument and mathematical induction it can be shown that the events(Aj)_j∈

are also independent, concluding the proof. N

The above lemma shows that outside a set of probability zero we can dene the right inverse αn of Vn 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 fol- lowing.

Proposition 3.2. The random walk (S_n)_n∈

N is a recurrent irreducible (F_n)_n∈N-martingale.

Proof. First note that by the denitions ofS_n andα_n, we have S_αn =· · ·=S_αn+1−16=S_αn+1

and

P S_αn+1−S_αn =±1

= 1 2, for alln∈N, so the time-changed random walk(S_αn)_n∈

Nis a symmetric simple random walk on Z.

By a theorem due to Pólya, the symmetric simple random walk onZ is recurrent (see for example [Bi], pp. 117118). This, together with the fact that by Lemma 3.1 the process Vn is a.s. increasing to innity (hence the same holds true for its inverse αn), shows that the random walk (Sn)_n∈N is also recurrent.

To prove the second claim, since Sn 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 :Sn=k)≥P(Si=i·sgn(k), i= 1, . . . ,|k|)

=P(Ui−1,i−1 = 1, Yi = sgn(k), i= 1, . . . ,|k|)

=p 2

k

>0, for any k∈Z^∗.

To prove the last claim, consider theσ-algebra

Fe_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 isFe_n-measurable, andY_n+1 is independent ofFe_n.

Using the properties of conditional expectation, we obtain E(S_n+1| F_n) =S_n+E(X_n+1| F_n)

=S_n+E E

U_Sn,nY_n+1|Fe_n F_n

=Sn+E

U_Sn,nE

Yn+1|Fe_n F_n

=S_n+E(U_Sn,nE(Y_n+1)| F_n)

=S_n+E(Y_n+1)E(U_Sn,n| F_n)

=Sn+ 0·E(USn,n| F_n) =Sn, concluding the proof.

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(Sn)_n∈N. (SLLN) Almost surely we have

(3.2) lim

n→∞

Sn

n = 0.

(CLT) We also have

(3.3) Sn

√np

→D Z (stably),

where Z is a standard normal random variable.

(FCLT) Moreover, if (ξ_n(t))_0≤t≤1 is the continuous process ξn(t) = 1

√np(S_k+X_k+1(nt−k)), k

n ≤t≤ k+ 1

n (k= 0,1, . . . , n−1), composed of the straight line segments joining the points

k n,^√S_np^k

0≤k≤n, then all nite dimensional distributions of ξ_n(t) converge weakly asn→ ∞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∈

Nis a symmetric simple random walk. By the classical SLLN (see for example [Bi, Theorem 6.1]) we have a.s.

(3.4) lim

n→∞

Sαn

n = 0.

Since by Lemma 3.1 the process Vn 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

n→∞lim Sα_Vn

V_n = 0.

Note that in the above it is possible that V_n = 0 for some indices n = 1,2, . . ., so the sequence ^SαVn_Vn is not dened for these indices n. However, by Lemma 3.1 Vn is a.s. increasing to innity, so the rst few terms of the sequence ^SαVn_Vn are not well dened only on a set of zero probability, which we ignore (alternately, we can dene ^SαVn_Vn = 0 if Vn= 0).

By the denition of the nondecreasing processVn and its inverseαnit 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

(3.5) lim

n→∞

Sn

Vn

= 0 a.s.

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∈

Nare also identically distributed (with mean E(US0,0) = p), using again the classical SLLN it follows that a.s. we have

(3.6) lim

n→∞

V_n

n = lim

n→∞

n−1

P

i=0

U_Si,i

n =EU_S0,0=p.

Using (3.5) and (3.6) we obtain

n→∞lim S_n

n = lim

n→∞

S_n Vn

· lim

n→∞

V_n

n = 0·p= 0 with probability 1, which concludes the rst part of the proof.

Next, note that since by Proposition 3.2S_nis aF_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, sn=p

Var (Sn),F_ni=F_i andSni =s⁻¹_n Sn. We have

s²_n= Var (Sn) =E

n

X

i=1

X_i²

!

=E

n

X

i=1

U_S²_i−1,i−1Y_i²

!

=E

n−1

X

i=0

USi,i

!

=np, so the corresponding martingale increments are given byX_ni=S_ni−Sn,i−1= s⁻¹_n Xi= ^√1_npXi,1≤i≤kn,n≥1.

In order to prove the last claim of the theorem, we will apply the Mar- tingale Central Limit Theorem (MCLT) to the martingale arraySni.

For an arbitrary ε >0 we have P

1≤i≤kmaxn

|X_ni|> ε

=P

1≤i≤kmaxn

U_Si−1,i−1Y_i > ε√

np

=

=P

1≤i≤kmax_nU_Si−1,i−1 > ε√ np

≤P

1≤i≤kmax_nU_Si−1,i−1 >1

= 0 for any n≥ _pε¹₂, which shows that

(3.7) max

1≤i≤k_n|X_ni|→^P 0.

By the rst part of the proof we have a.s. lim

n→∞

1 n

n−1

P

i=0

U_Si,i = p, and therefore we obtain

n→∞lim

kn

X

i=1

X_ni² = lim

n→∞

1 np

n

X

i=1

U_S²_i−1,i−1Y_i² = lim

n→∞

1 np

n−1

X

i=0

USi−1,i−1 = 1 a.s., and in particular

(3.8)

kn

X

i=1

X_ni² →^P 1.

Finally, for alln≥1 we also have E

1≤i≤kmax_nX_ni²

= 1 npE

1≤i≤kmax_nU_S²_i−1,i−1Y_i²

= 1 npE

1≤i≤kmax_nU_Si−1,i−1

≤ 1 np,

which shows that

(3.9) E

1≤i≤kmaxn

X_ni²

is bounded inn.

The relations (3.7)(3.9) 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, Sn is a F_n-martingale with S₀= 0, and for anyn≥1 we have

σ²_n=E X_n² F_n−1

=E USn−1,n−1

F_n−1

=X

k∈Z

E Uk,n−11_{Sn−1=k}

F_n−1

=X

k∈Z

1_{Sn−1=k}E(Uk,n−1| F_n−1)

=X

k∈Z

1_{Sn−1=k}E(Uk,n−1)

=pX

k∈Z

1{Sn−1=k} =p.

It follows that Pⁿ

i=1

σ²_i =np ands²_n=E _n

P

i=1

σ²_i

=np, so trivially

(3.10)

n

P

i=1

σ_i² E

_n P

i=1

σ_i²

= 1−→^P 1.

Also note that 1

s²_n

n

X

i=1

E X_i²1_{{|Xi|≥εsn}}

= 1 np

n

X

i=1

E

U_S²_i−1,i−1Y_i²1ⁿ

U_Si−1,i−1≥ε√ np

o

= 1 np

n

X

i=1

E

U_Si−1,i−11ⁿ

U_Si−1,i−1≥ε√ npo

= 0

for all n > _ε¹2p (sinceUi,j ∈ {0,1}for alli∈Z,j ∈N). In particular, it follows that the Lindeberg condition holds for Sn, that is we have

(3.11) 1

s²_n

n

X

i=1

E X_i²1_{{|Xi|≥εsn}} P

−→0.

The last claim of the theorem follows now from MFCLT (see for example [Br, Theorem 2]) using (3.10)(3.11), 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 time by a factor of ¹_n, space by ^√1_np

, and for large values of n, the probabilities concerning the random walkS_ncan 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 individ- uals. 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.

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/62988.

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Received 9 August 2012 Transilvania University of Bra³ov Faculty of Mathematics and Computer Science Str. Iuliu Maniu Nr. 50, 500091 Bra³ov, Romania

mihai.pascu@unitbv.ro and

Simion Stoilow Institute of Mathematics of the Romanian Academy

P.O. Box 1-764, 014700 Bucharest, Romania