Abstract
In the model of choice, studied in this paper, the decision maker chooses the actions non-probabilistically in each period (Sarin and Vahid, 1999; Sarin, 2000) .
Abstract
The action is chosen if it yields the biggest payoff according to the decision maker's subjective assessment. Decision maker knows nothing about the process that generates the payoffs. If the decision maker remembers only recent payoffs, she converges to the maximin action. If she remembers all past payoffs, the maximal expected payoff action is chosen. These results hold for any possible dynamics of weights and are robust against the mistakes. The estimates of the rate of convergence reveal that in some important cases the convergence to the asymptotic behavior can take extremely long time. The model suggests simple experimental test of the way people memorize past experiences: if any weighted procedure is actually involved, it can possibly generate only two distinct modes of behavior.
Abstract
JEL classification: D83.
Introduction
The environments in which economic agents make decisions can be very complex. This accounts for the tendency of the agents to simplify their decisions. I study the behavior of a simple decision maker who does not randomize while choosing among actions. Instead, she chooses the action with the biggest subjective assessment, which is represented by the weighted average of the past payoffs. The decision maker has no information about the environment apart from the payoffs she receives. I analyze the long run behavior of the decision maker and the rate of convergence for very general weight structure.
Introduction
The model of the kind was first introduced in Sarin and Vahid (1999) . In this paper the decision maker does not randomize among actions as well, but the weights given to the past payoffs are fixed. The long run behavior of the decision maker depends on various assumptions and might be stochastic (i.e. in the long run the agent is randomizing among several actions).
Introduction
In this paper I consider a model which makes sharp predictions about the long run behavior for all possible weight configurations. Depending on the "size" of the memory, the decision maker can follow only two modes of behavior. Moreover, the result is robust against the mistakes by the decision maker. This suggests an easy way to test if the actual mechanism of memorizing payoffs in humans works in weighted average fashion.
Introduction
An experimental study in Sarin and Vahid (2001) already shows that the models of this type can explain existing data exceptionally well. The only drawback that becomes clear is that the rate of convergence to the long run action might be tremendously low in some cases. This can potentially create problems with testing the hypotheses.
Introduction
In the model the decision maker faces the same decision problem repeatedly. In each period she chooses an action according to her subjective assessments, which are the weighted averages of the past payoffs. After the choice is made, the state of the world is realized and the payoff is revealed. The decision maker has no information about the process that generates the payoffs. After the payoff is known, she updates the subjective assessment of the action which generated the payoff. The subjective assessments of all other actions stay the same. We assume that the decision maker can make mistakes with ε probability in which case she chooses an action with non-maximal assessment.
Introduction
Similar model is studied extensively in computer science literature, where it is called ε-greedy learning procedure (Sutton and Barto, 1998) . Most results in computer science, however, are simulations that compare this learning model to others. Very close learning model is used by Young (2007) to study the behavior in large games. When the action sets and the number of players are very big it becomes computationally impossible for either humans or computer agents to use learning procedures that try to take into account the information about the game. Thus, it is important to understand how learning models that use only payoff information behave.
Introduction
The paper is organized as follows. In section 2 the model with finite memory is formalized and the maximin result is proved. General model is introduced in section 3 as well as the long run behavior results for different cases. Section 4 deals with the estimates of the rate of convergence. And Section 5 concludes. Proofs and definitions can be found in the Appendix 6.
The Model with Finite Memory
The decision maker faces the same decision problem repeatedly. Each period she is choosing one of the J actions from a finite set A = {a 1 , a 2 , ..., a J }. After the action is chosen, the state of the world ω t ∈ Ω is realized (t stands for the time period). The set of states of the world Ω is assumed finite. Suppose that there is some probability measure defined on Ω. For each t, ω t is identically and independently distributed in each time period. After the state of the world is chosen decision maker receives her payoff according to the utility function u : A × Ω → R + (in what follows I will write u i (t) for the payoff received by the decision maker from choosing ith action in period t). Decision maker does not have any information about the process that generates the payoffs (even if it is random or not).
The Model with Finite Memory
To choose an action the decision maker uses the following information. There exist real-valued subjective assessments of the payoffs, which the decision maker attaches to each action in A. Denote these assessments by α i (t) for ith action in period t and write the vector of assessments as α(t) = (α 1 (t), α 2 (t), ..., α J (t)). Each period the decision maker chooses the action with the maximal subjective assessment. For simplicity, assume that at any time t all the subjective assessments are different from each other. This assumption does not create any loss of generality but helps to avoid uninteresting ties.
The Model with Finite Memory
Subjective assessments are updated each period. The assessment for each action i at period t is a time and action invariant function of the payoffs, received by the decision maker after playing i in the past. These payoffs are stored in the memory. Assume that the memory of the decision maker is finite and she remembers m values of the payoffs received in the past from each action. Denote by m i (t) the "memory vectors" for each action i at time t. These vectors contain the values of last m payoffs, which where obtained from playing each action. So m ′ i (t) = (u i (t 1 ), u i (t 2 ), ..., u i (t m )), where t 1 < t 2 < ... < t m ≤ t are the periods when action i was played last m times in the past (the prime means transposition). The memory vector for an action changes whenever the action is played. The first (the oldest) value of the payoff in the memory vector disappears (the decision maker forgets it), all other values in the vector move one position leftward and new payoff takes its place in the rightmost position. For example, if the decision maker has played action i and received some payoff u i (t) her memory for action i changes as follows:
The Model with Finite Memory
are the m − 1 values taken from the vector m i (t). Call this transformation of memory vector T (m i (t), u i (t)). Think of m i (0) as given for each action i and that all elements in m i (0) are equal to one of the payoffs, say u i (0), which action i can possibly yield. Let us then assume that α i (0) = u i (0). Now consider how the subjective assessments are calculated. Suppose that there exist the constant vector of non-negative weights λ ′ = (λ 1 , λ 2 , ..., λ m ) with the property m i=1 λ i = 1. The assessment of the payoff from playing ith action in period t is the λ-weighted average of the values of previous payoffs received from playing action i. More formally, in the end of the period t, when the decision maker obtains the payoff u i (t) from choosing ith action, the memory and the assessments change as follows:
The Model with Finite Memory
Only the memory and the assessment of the action that was played changes. All other memory vectors and assessments remain the same.
The Model with Finite Memory
In addition, assume that the decision maker can make mistakes while choosing the action to play. Each period she chooses the action with maximal subjective assessment with probability 1 − ε, where ε is some small positive number. With probability ε she makes a mistake and some other action is chosen instead. So, with probability 1 − ε the decision maker chooses her subjectively maximal action and all other actions in A are chosen with probability ε/(J − 1) each.
The Model with Finite Memory
Say that the decision maker has finite memory if she remembers only m values of the payoffs received in the past (for each action). This is equivalent to saying that the memory can be in only finite number of states. Indeed, denote the set, which contains 
The Maximin Result
In this section I will prove the result concerning long-run properties of the model above.
The Maximin Result
In particular, I will show that in the long run the decision maker will choose the maximin action (i.e. the action with maximal minimum payoff) almost surely. The proof will proceed in two stages. First, the model without mistakes, i.e. with ε = 0, will be considered and it will be shown show that it can be described as a discrete-time Markov process with finite state space. Second, I will show that the original model with ε > 0 is a regular perturbed Markov process which converges in some sense to the case ε = 0 as ε gets small.
The Maximin Result
Call the model with ε = 0 the unperturbed system. The state of the system is fully described by the state of the memory of the decision maker. Indeed, given some state of the memory µ ∈ M decision maker's subjective assessments are determined unambiguously. So, the probabilities of getting to other states are determined without ambiguity and depend only on the probability measure over Ω.
The Maximin Result
Definition 2.1 Denote by P M,λ,0 the discrete-time Markov process with finite state space M , which fully describes the unperturbed system.
The Maximin Result
Proposition 2.2 In the unperturbed system the decision maker chooses only one action asymptotically. This action is the maximin action in A (the one with maximal smallest payoff ).
The Maximin Result
Proof. Given our assumption that the assessments in period 0 are in between maximal and minimal payoff for each action the proof of Proposition 1 in Sarin and Vahid (1999) gives the result.
The Maximin Result
The idea of the proof is as follows. The assessments of the payoffs from each action can only lie in between action specific maximal and minimal payoffs. Thus, whenever the assessment of some not maximin action falls below maximin that action is never chosen again. With time the assessments of all actions go down. This happens since the decision maker switches to some other action only when the assessment of currently played action falls below some other assessment, then the process repeats itself. As time passes all assessments will become lower than maximin. Now as we found the action, which is chosen by the decision maker in the unperturbed system, let us make some statements about Markov process P M,λ,0 . In the following Proposition assume without loss of generality that the maximin action has index 1 (a 1 ).
The Maximin Result
Also denote the minimal payoff from action a i by u i min = min ω∈Ω u(i, ω). Denote the maximin payoff by u max min = max 1≤i≤J u i min .
The Maximin Result
Proposition 2.3 The Markov process P M,λ,0 has finitely many recurrent classes. All of these recurrent classes consist only of the states in which the decision maker chooses maximin action. The set of states in any recurrent class can be described as
The Maximin Result
.., J} is some constant value of the collection of J − 1 memory vectors corresponding to not maximin actions with the property that the assessments of these actions (also constants) are less than maximin payoff.
The Maximin Result
Proof. See Appendix 6.1.
The Maximin Result
Now the analysis of the behavior of the model with mistakes is possible. Recall that in each period the decision maker chooses the action with maximal subjective assessment with probability 1 − ε and any other action is chosen with probability ε/(J − 1).
The Maximin Result
It is easy to see that the behavior of the model with mistakes is described by the Markov process similar to the model without mistakes. The difference is that from any given state µ ∈ M the process can get to the states previously unreachable. The probability of these events is proportional to ε. Say that the Markov process P M,λ,ε is a perturbation of the process P M,λ,0 if the transition matrix of P M,λ,ε is slightly dis-torted version of the transition matrix of P M,λ,0 . Suppose that P M,λ,ε fully describes the behavior of the model with mistakes.
The Maximin Result
Definition 2.4 (Young, 1998) The perturbed Markov process P M,λ,ε is called regular perturbed Markov process if it satisfies the following conditions: 1
The Maximin Result
2. for every two states z, z ′ ∈ M it is true that
The Maximin Result
Proposition 2.5 The perturbed Markov process P M,λ,ε is a regular perturbed Markov process.
The Maximin Result
Proof. See Appendix 6.1. Now, according to the Theorem 3.1 of Young (1998) any regular perturbed Markov process P M,λ,ε has unique stationary distribution µ ε for each ε > 0. More than that, lim ε→0 µ ε = µ 0 , where µ 0 is some stationary distribution of the unperturbed process P M,λ,0 . So as ε → 0 the perturbed system converges to some stationary distribution of the unperturbed system. The stationary distribution of the unperturbed system necessarily lies in one of the recurrent classes. Since we have shown that in all the states inside any recurrent class the decision maker chooses maximin action, the same maximin action will be chosen in the perturbed system with probability close to one. When ε > 0 is small enough the decision maker will "circulate" around maximin action, but still choose it most of the time.
General Model
In this section I consider the case when the memory of the decision maker is infinite.
General Model
This will be the only difference of the model in this section from the model with finite memory. In each period the decision maker chooses the action with maximal assessment.
General Model
At the same time the decision maker can make mistakes and choose some different action instead. There are three different modes of behavior which can emerge when the memory is infinite. Difference arises because of the way of formation of weights λ i in the infinity.
General Model
All the assumptions of the model with finite memory remain the same, only the assessments updating rule changes. Here I do not assume that before choosing an action for the first time the decision maker has already infinite number of payoff experiences.
General Model
Instead, suppose that in the beginning the decision maker has some given vector of assessments α(0) = (α 1 (0), ..., α J (0)), with each component being between the minimum and the maximum possible payoff from each action. After receiving the payoff from playing action a i the decision maker updates her assessments in the following way:
General Model
Here λ t = (λ tt , λ t(t−1) , ..., λ t0 ) ′ denotes the column vector of length t + 1 of the weights attached to the payoff just received and the payoffs from playing the same action in the past. The column vector m ∞ i (t) of length t + 1 contains the payoff just received in the period t, all the payoffs received by the decision maker from playing action a i in the past, and the value of α i (0):
General Model
As before assume that the decision maker finds weighted average of the payoffs, so t i=0 λ ti = 1 ∀t ∈ N. As time passes the vectors become longer as more and more payoffs are put to the memory of the decision maker. There are no assumptions on the particular way of evolution of weights as more payoffs are received. The only restriction is that there is some procedure, the same for all actions, which generates the weights each period. 2 Now as the model is described, let us define the three different ways the weights λ can behave in infinity. Consider the triangular array of weights 1 λ 11 λ 10
General Model
where t i=0 λ ti = 1, ∀t ∈ N. In each period t the most recent payoff received by the decision maker is given the weight λ t0 . The oldest payoff (α i (0)) is given the weight λ tt . Consider three types of arrays.
General Model
Case I. This case is close to the finite memory model as no conditions are imposed on weights to stay bounded away from zero. For any triangular array of weights λ define a function N λ : N × R → N by
General Model
Say that the triangular array of weights λ belongs to Case I whenever
General Model
Case II. In this case the decision maker takes into account infinitely many past payoffs.
General Model
As t → ∞ all the weights approach zero, however the weight does not "escape" to infinity. Case III. In this case the weight "escapes" to infinity. Any array of weights which does not satisfy No Escape Condition belongs to Case III.
Long Run Behavior in the General Model
In this section I prove results concerning long run behavior of the decision maker as the number of periods played grows to infinity. Depending on the way the weights are formed, the decision maker's behavior differs in the long run. Proof. See Appendix 6.1.
Long Run Behavior in the General Model
It can be easily seen that when we introduce mistakes to the model with Case I weights we still get robust convergence to maximin action (in the sense of Section 2.1).
Long Run Behavior in the General Model
In the special case when the array of weights converges row-wise to some element of the space ℓ 1 we can get as good approximation of infinite memory by finite memory models as we desire since the convergence to maximin result holds for any finite memory model. 3
Long Run Behavior in the General Model
If the array of weights does not converge in ℓ 1 we still can approximate by finding the finite sequence of weights (λ 1 , ..., λ n ) with the function N (n, δ) "close" to the function of the array as n → ∞.
Long Run Behavior in the General Model
Proposition 3.3 In the model with mistakes and Case II weights the decision maker asymptotically plays the action with maximal expected payoff with probability 1 − ε.
Long Run Behavior in the General Model
Proof. See Appendix 6.1.
Long Run Behavior in the General Model
In Case III nothing specific can be said about the long run behavior of the decision maker since asymptotically she puts positive weights only on the payoffs received infinitely long time ago. Depending on the realizations of payoffs anything can happen.
Long Run Behavior in the General Model
However, this case can still have some meaning to it. For example, if the decision maker has very strong "first impression": after playing some action for the first time she remembers only the first payoff and never changes her assessment afterwards. This behavior corresponds to the array of weights with λ nn = 1 ∀n ∈ N and λ nk = 0 ∀k < n 3 ℓ 1 is the space of all infinite real sequences (x 1 , x 2 , ...) with the norm ∞ i=1 |x i |.
Rate of Convergence
In this subsection I will state the results concerning the rate of convergence to maximin in the finite memory model and to maximal expected payoff in Case II model.
Finite Memory Model
It seems intuitively true that bigger memory slows down the convergence to maximin.
Finite Memory Model
Recall the proof of Proposition 2.2. There it was stated that if the decision maker chooses non-maximin action then in finite number of steps, the subjective assessment of this action would fall to some level below the value of maximin payoff. The subsequent occurrence of m lowest payoffs, for example, is enough. If m is very large, then the subjective assessment will be an average of a big number of random variables and it will take longer for it to fall lower than the maximin payoff. Thus, as m grows it will take longer and longer for the decision maker to switch to the maximin action.
Finite Memory Model
I find an estimate of the minimum number of periods needed for the decision maker to converge to maximin action. Say that the decision maker has converged to maximin action once the subjective assessments of all non-maximin actions become less then the value of maximin payoff for the first time. This means that "the convergence takes place" whenever the decision maker chooses maximin action not erroneously. Of course, since the decision maker makes mistakes, it can happen that she will switch to some nonmaximin action in the future. However, since the assessments of all other non-maximin actions will still be less than the value of maximin payoff, the decision maker will switch back to the maximin relatively quickly (especially if the number of actions is big).
Finite Memory Model
Denote by F i the distribution function of the subjective assessment α i , which is a weighted average of m discrete random variables. Without loss of generality let us assume that maximin action is the action a 1 with corresponding subjective assessment α 1 .
Finite Memory Model
Consider the sequence of periods t j 1 , t j 2 , ... when action a j is chosen. We can treat the values of the subjective assessment α j in periods t j 1 , t j 2 , ... as a sequence of independent realizations of random variable with distribution function F j . 4 Each period the action a j is chosen the probability that the subjective assessment of this action will become less than u max min equals to P j = F j (u max min ). Simple calculation shows that the expected number of periods needed for the subjective assessment α j to become less than u max min
Finite Memory Model
Let us think about the sequences of periods when each action is played as a separate sequence. We need to find the minimum expected time needed for all non-maximin assessments to become less than u max min . The expected number of periods, which is necessary for this, is the sum of the expected number of periods for each action:
Finite Memory Model
here we use our assumption that the maximin action has index 1.
Finite Memory Model
In general, N depends on 1) distributions of all u i except the distribution of maximin action; 2) the value of u max min ; 3) number of actions J; 4) length of memory m (F i depend on length of memory, the longer is the memory, the smaller is the variance of the distributions). It is straightforward that N is an increasing function of m and J and a decreasing function of u max min .
Case II Model
Another approach is used here to estimate the number of periods needed for convergence to maximal expected payoff. Denote by V i the variance of the random variable u(i, ·), by E i its expectation and let σ i = √ V i . Without loss of generality assume that action 1 has the biggest maximal payoff. Let ∆ i = E 1 − E i . I am interested in how the variance of the assessments behaves. Notice that given any assessment which was updated t times the variance of the assessment satisfies
Case II Model
Denote by σ it the standard deviation of assessment α i after t updates, then
Case II Model
The question now is what is the number n of updates such that for some κ the intervals [E i , E i + κσ in ] and [E 1 − κσ 1n , E 1 ] do not intersect. The intuition is that if these two intervals are disjoint then the assessment i is less than assessment 1 with probability at least 1 − 1/κ 2 (Chebyshev's inequality). Clearly the minimal number of updates solves
Case II Model
Suppose that n * i solves this equation for each i. Then the number of updates needed for the assessments of all actions to be less than the assessment of action 1 with proba-
Case II Model
The max i>1 n * i term is needed as action 1 should be chosen enough times as well.
Case II Model
It is not easy to explicitly find n * i in general, however we can find an estimate of N which involves only min k λ nk and max k λ nk by using the inequality above. True n * i lies in between the solutions to
Case II Model
Suppose that n U i solves the first equation and n L i the second. Then
Case II Model
For example, assume that min k λ nk = 1/n µ where 0 < µ ≤ 1 then the minimal number of periods needed for the action 1 to be played with probability
Case II Model
One can see that the rate of convergence can be tremendously low if µ is small (say, µ = 0.1). This means that if the decision maker uses the weights that go to zero not too fast, but still fast enough for the assessments to converge to expectation, then we should not expect to see the maximal expectation action to be chosen.
Conclusion
Economic agents tend to simplify their decisions when facing very complex environment.
Conclusion
The model in this paper studies the behavior of a simple decision maker, who does not possess any information about the process that generates payoffs. The decision maker simplifies her choice by choosing actions non-probabilistically.
Conclusion
The results differ depending on the weights the decision maker attaches to the past payoffs. If she cares only about recent payoffs then the maximin action is chosen in the long run. This result holds even if we introduce mistakes into the model. If the decision maker cares about payoffs received long time ago then she converges to the maximal expected payoff action. In the maximin case, the rate of convergence to the long run behavior depends on the number of actions, length of the memory and the process that generates the payoffs. In the maximal expectation case, the rate of convergence depends on how fast the biggest weight goes to zero. Remarkably, this rate can be very low in some cases when the weights decrease slow enough. Huck and Sarin (2004) ; Young (2007) ).
Conclusion
6 Appendix 6.1 Proofs Proof of Proposition 2.3. The recurrent class of the Markov process is a set of the states, which satisfy two properties: 1) any pair of states in the recurrent class communicate; and 2) none of the states outside the recurrent class can be accessed from any of the states inside it. To prove property 1 of this statement notice that once the decision maker chooses maximin strategy and α i < u max min for all i = 2, ..., J she chooses only it forever after. This means that, after it happens, the memories (and the assessments) of all not maximin actions are not updated anymore and are constant forever (this constant is denoted Q in the Proposition). However, the decision maker continues to choose maximin action and so its memory vector is changing all the time.
Conclusion
Given that all other memory vectors are constant, it is easy to see that the decision maker can reach any state of the maximin memory vector. 2m steps is enough to reach any memory configuration from any other. Any of these 2m steps has positive probability of occurrence. Thus any set of states described in the Proposition satisfies property 1. Property 2 follows from the proof of the Proposition 2.2. So we conclude that any set of states of the form stated in the Proposition constitutes a recurrent class of the Markov process P M,λ,0 and all of these recurrent classes consist only of the states in which the decision maker chooses maximin action. There is no other set of states, which do not belong to some of the recurrent classes described in the Proposition, and, at the same time, constitute a recurrent class. This again directly follows from the proof of the Proposition 2.2.
Conclusion
Proof of Proposition 2.5. We first prove that P M,λ,ε is irreducible. It is enough to show that any two states z, z ′ ∈ M communicate. Consider any two such states z and z ′ . There exists a sequence of mistakes together with particular states of the world occurring at each transition such that with positive probability the system will move from state z to z ′ . It is easy to imagine such sequence which gradually makes the same all memory vectors in z and z ′ . Now we show that conditions 2 and 3 of Definition 2.4 are satisfied. Indeed, in all possible perturbations in the transition matrix, ε enters as a linear term cε where c is some constant. This is enough for condition 2 to be true. For the same reason the condition 3 is satisfied when r(z, z ′ ) = 1 for any z and z ′ such that P M,λ,0 zz ′ = 0 and when r(z, z ′ ) = 0 for any z and z ′ such that P M,λ,0
Conclusion
Proof of Proposition 3.2. Given the definition of the Case I array of weights we can be sure that for any given δ > 0 we can find a number n δ such that
Conclusion
Since all u i min are different, this guarantees that there exists δ small enough for any assessment to fall below u max min with positive probability (n δ occurrences of u i min will suffice). Now we get the result by placing this instead of the analogous argument in Proposition 1 of Sarin and Vahid (1999) .
Conclusion
Proof of Proposition 3.3. Let us show that the assessments of all actions converge to the expected value of payoffs. Consider some action i and time periods t 1 , t 2 , ... when this action is played (t 1 < t 2 < ...). We are interested in the behavior of the assessment α i (t n ) as n → ∞ (since the decision maker makes mistakes we can be sure that any action is played infinitely often). For each update period t n we know that
Conclusion
where u(i, ω t ) is the realization of the payoff in period t. To prove the statement use the theorem from Fristedt and Gray (1997) (Theorem 25, p.311). 5 First we show that the assumptions of the theorem are satisfied and then explicitly find the limiting distribution of α i (t).
Conclusion
Consider the triangular array of random variables:
Conclusion
. . . λ nn α i (0) λ n(n−1) u(i, ω t 1 ) . . . λ n0 u(i, ω tn )
Conclusion
. . . 5 See Appendix 6.2
Conclusion
This random variable has moment generating function exp(−E[u(i, ·)]) which obviously corresponds to delta distribution at E[u(i, ·)]. This finishes the proof of the statement above.
Conclusion
In the model the decision maker makes mistakes, so all the actions are played infinitely often. Therefore, as it was shown, assessments of all actions converge to the expected value. Since the decision maker chooses the action with the maximal assessment, she will eventually choose the one with maximal expected payoff.
Triangular array problem
Consider the triangular array of random variables 7 X 11 X 22 X 21 X 33 X 32 X 31 . . .
Triangular array problem
X nn X n(n−1) . . . X n1
Triangular array problem
. . .
Triangular array problem
For each n the vector (X nk : k = 1, ..., n) is independent. So call such an array row-wise independent. How do the row sums S n = n k=1 X nk which come from the uan arrays behave? The following theorems characterize this.
Triangular array problem
A measure ν on (0, ∞) is a Lévy measure for R + if (1 − e −vy )ν(dy) If X 1 and X 2 are random variables with distributions Q 1 and Q 2 let Q 1 * Q 2 mean the distribution of X 1 + X 2 and i Q i be the sum of distributions as measures, so that Q 1 + Q 2 is a measure that can take on values up to 2. Theorem 6.3 (Fristedt, Theorem 25, p.311 ) Let (Q nk : 1 ≤ k ≤ n, n = 1, 2, ...) be a uan triangular array of distributions on R + . For each n, let Q n = Q n1 * Q n2 * ... * Q nn In order that the sequence (Q n : n = 1, 2, ...) converge to a distribution on R + it is necessary and sufficient that there exist a nonnegative number ξ and a Lévy measure ν for R + satisfying the following two conditions: x Q nk (dx)
Triangular array problem
In case these conditions are satisfied, the sequence (Q n : n = 1, 2, ...) converges to the infinitely divisible distribution on R + corresponding to (ξ, ν) via the Lévy-Khinchin Representation Theorem for R + .

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