ON THE TAIL BEHAVIOR OF FUNCTIONS OF RANDOM VARIABLES

It is shown that the tail behavior of the function of nonnegative random variables can be characterized using deterministic functions satisfying certain properties. Also, the upper and lower bounds for the tail of product of random variables are given. Applications of these results are given to some of the well-known models in economics and risk theory. AMS Subject Classification: 60E05, 60E15, 62P20, 91B30, 91B40


Introduction
The tail behavior of functions of random variables (rvs) is an important area of research.For theoretical development on this topic, see [11,8,2] and references therein.In applied probability, some of the branches that rely on the analysis of the stochastic model, described by given function(s) of rvs, it is important to estimate the behavior for the given function(s) of rvs.For example, in reliability theory, the tail behavior of failure distribution plays an important role (see [1,7]).In risk modelling, the behavior of the distribution of ruin, for risk model is important (see [9]).In this paper, we focus on some aspects of the tail behavior and generalize the existing results for various functions of rvs, such as sum, maximum and product under dependent and independent setup.Also, the result for moment and exponential indices follows as a special case of our results, provided they exist.
We next define the necessary terminology required for the discussion.Let X be the rv with cumulative distribution function (cdf) F X (x).Then FX (x) = 1− F X (x) known as the tail function of X.The moment and exponential index, (see [4] and [5]), for a rv X are defined as respectively, where x + = max(0, x) and R X (x) = − ln FX (x) , the hazard function of X.A function h : [0, ∞) → [0, ∞) such that h is increasing and h(x) → ∞ as x → ∞ known as a scale function.If h is continuous then we can generalize the definitions of moment and exponential index (see Theorem 2.1 of [6]) as follows.
is called the h-order of X and h is said to be natural scale function if I h (X) = 1.Using this definition, the tail of the rv X can be compared as, for ǫ > 0, FX (x) ≤ e −(1−ǫ)h(x) , and for the rvs X and Y , if lim inf x→∞ R X (x)/R Y (x) = c and c > 0. Then the tail comparison (for details, see [6]) is given by, for any small ǫ > 0, there exists Next, we introduce a result of the existence of h satisfying the required properties for discussing the tail behavior of a nonnegative rv.Lemma 1.1.Let X be nonnegative rv.Then there exists a monotone concave function Proof.It is clear that X is either heavy-tailed or light-tailed rv.For heavytailed rv X, Theorem 2.9 of [11] gives the required result.For light-tailed rvs, we can take h(x) = x α for any 0 < α < 1 is a monotone concave function satisfying h(x) = o(x) and Ee h(X) < ∞.
In this paper, we consider nonnegative continuous rvs with right unbounded support, that is, for a rv X, P(X > c) > 0 for all c > 0. The structure of the paper is as follows.In Section 2, we first prove the theorem to characterize the tail behavior of functions of (dependent or independent) rvs, such as sum, maximum and product.Next, we derive a method to find the natural scale function for differentiable functions of rvs and use this method to prove the theorem which gives the bounds for the tail of rv XY .Finally, in Section 3, we apply the results of Section 2 to Cobb-Douglas production model and discrete time risk model.

The Moment Index for Sum, Maximum and Product of Random Variables
In this section, we obtain some of the results about the tail behavior of functions of rvs based on the h-order defined in (1).We exemplify the approach of Lemma 1.1 to various functions of rvs such as sum, maximum and product in the following results.
Theorem 2.1.Let X 1 , . . ., X n be nonnegative rvs.Then there exists a monotone concave function h and 0 < c 1 ≤ c 2 ≤ 1 such that Theorem 2.2.Let X 1 , . . ., X n be nonnegative rvs.Then there exist functions h and ĥ such that Corollaries 2.1.Let X 1 , . . ., X n be independent nonnegative rvs.Then there exists a monotone concave function h such that 1. Observe that, if h satisfies h ( n i=1 x i ) ≤ n i=1 h(x i ), the condition of concavity can also be relaxed (see Theorem 4 of [6]).

If
3. Observe that, for X = 0 and h(0) = 0, applying (1), we get I h (0) = sup{s ≥ 0 : E e sh(0) < ∞} = sup{s ≥ 0 : , our results for the case of exponential and moment indices follows immediately provided they exist.However, our results give flexibility for the choice of h i 's.

Natural Scale Function
Recall from (1) that, a scale function h is called natural scale function for a rv X if I h (X) = 1.Next, we describe a method to find the natural scale function for functions of rvs via transformation technique.
Method.Let X 1 , . . ., X n be continuous rvs with support S. In particular, assume Y 1 = g(X 1 , . . ., X n ) is a differentiable function of n rvs.Then, the problem is to find the natural scale function for Y 1 .Consider an integral where R ⊂ R n .Now, take the transformation of the form y 1 = g(x 1 , . . ., x n ) and y i = x i for i = 2, 3, . . ., n with support T , together with inverse functions x 1 = w(y 1 , . . ., y n ) and x i = y i for i = 2, 3, . . ., n.Then, using transformation technique (see [10], pp-124), it is well-known that the joint pdf of rvs Y where |J| is the determinant of the Jacobian matrix J. Also, the marginal distribution of Y 1 is |J|f X 1 ,...,Xn (w(y 1 , . . ., y n ), y 2 , . . ., y n ) dy 2 . . .dy n and the tail function is given by FY We can compare the tail of the random variable . In particular, we have shown that for any differentiable function of g(X 1 , . . ., X n ), we can find a natural scale function h, such that, for any ǫ > 0, FY 1 (y 1 ) (the tail function of g) is dominated by e −(1−ǫ)h(y 1 ) .
Next, we present the upper and lower bounds for the product of two iid rvs.
Theorem 2.3.Let X and Y be nonnegative iid rvs.Then, for small ǫ > 0, we have the following inequalities, for x > 1, Remark 2.1.In the proof of Theorem 2.3, the natural scale function h defined in Case 1 and Case 2 are different.In Case 1, it is twice the natural scale function of X and in Case 2, it is the natural scale function of XY .

Applications
In this section, we give applications of our results to some well-known models in economics and risk theory.

Cobb-Douglas Production Model
In economics, the Cobb-Douglas production model describes the relationship between the output and input variables.This has been widely used since its introduction by Knut Wicksell .
The first significant application of this model is given in [3], where they studied the growth of American economy during the period 1899-1922 using this model, and they were able to present a simplified view of the economy in which the production output is determined by the amount of labor involved and capital invested.
It is also of importance in economics, to study the long-term behavior of production function in order to formulate certain policies.We next describe the mathematical formulation of the Cobb-Douglas model.
Let T (X, Y ) = bX α Y β , where T =total production (the monetary value of all goods produced in a year): X =labor involved (the total number of person-hours worked in a year), Y =capital invested (the monetary worth of all machinery, equipment, and buildings), b =total factor productivity, α > 0, β > 0 are the output elasticities of labor and capital, respectively.These values are constants determined by available technology.
We next give the implications of our results to Cobb-Douglas model.Now, consider the following conditions on α and β, for a detailed analysis.
and the natural scale function of Hence the tail of the production function dominated by the above function for α, β > 0 and α = β.That is, all three cases (Increasing return to scale, constant return to scale, decreasing return to scale) whenever α = β tail of the production function dominated by the above function.Now, consider the case when α = β = 1 and Theorem 2.3 for the Pareto distribution.
Example 3.2.Suppose X and Y are iid Pareto distributed rv with pdf given by f (x) = ak a x a+1 where a > 0 and k ≤ x < ∞.Now using technique given in 2.1 with Hence, it is easy to see that It is clear that Similarly, I h XY (Y ) = 1, hence for c = 1, it is satisfied Theorem 2.1.
Compare with production function of Cobb-Douglas model, we have α = β = b = 1.That is, T (X, Y ) = XY , i.e., for x > 1, we have the case in which increasing return to scale, From Theorem 2.3, the tail function of XY and X are dominated by the function,

Discrete Time Risk Model
Let X i be the net payout of the insurer at year i, and Y i be the discount factor (from year i to i− 1) related to the return on the investment, i = 1, 2, . . . .Then the discounted value of the total risk amount accumulated till the end of year n can be modeled by a discrete time stochastic process The basic assumptions for this model (see [9]) are as follows.
A 2 .Let r n be the rate of interest on the reserve invested is risky assets.Assume {r n : n = 1, 2, . . .} is a sequence of iid rvs with support (−1, ∞).
Define B n = 1 + r n , also known as the inflation coefficient from year n − 1 to year n and let Y n = B −1 n be the discount factor from year n to year n − 1, n = 1, 2, . . . .The rvs X = −A and Y as the insurance risk and financial risk, respectively.Clearly, P(0 < Y < ∞) = 1.
Let the initial capital of the insurer be x ≥ 0. The surplus of the insurer accumulated till the end of year n can be characterized by S n which satisfies the following recurrence equation where B n = 1 + r n , n = 1, 2, . . . .The recurrence relation can be written as where n j=n+1 = 1.Next, the time of ruin for the risk model ( 4) is defined as τ (x) = inf{n = 1, 2, . . .: S n < 0|S 0 = x}.Hence, the finite time ruin probability, ψ(x, n), and of ultimate ruin probability, ψ(x), can be defined as ψ(x, n) = P(τ (x) ≤ n), respectively, ψ(x) = ψ(x, ∞) = P(τ (x) < ∞).Clearly, the probability that the ruin occurs exactly at year n, can be defined as A more significant calculation might be P(τ y (x) ≤ n) or P(τ y (x) < ∞) for x > 0 and n = 1, 2, . . ., where τ y (x) is a stopping time, defined by τ y (x) = inf{n = 1, 2, . . .: S n ≤ y|S 0 = x} for any regulatory or trigger boundary y ≥ 0. This stopping time τ y (x) may be interpreted as the first time at which there is a need to raise the capital in order to maintain solvency.We can rewrite the discounted value of the surplus S n in as Hence, for each n = 0, 1, 2, . . ., ψ(x, n) = P(U n > x), where Hence, the following result shows that the relation ψ(x, n) = P(V n > x) holds for each n = 1, 2, . . .under the assumptions A 1 , A 2 and A 3 .
We next apply our results to discuss the tail behavior of U n .Using Theorem 2.1, Theorem 2.2 and Corollary 2.1, there exists a monotone concave function h such that with assumption that c 0 = 1, combining ( 5) and ( 6), we get where 0 Then, we can write Combining ( 7) and ( 8), we get Now, again take constants corresponding to ( i j=1 Y j )Y i+1 is d i , i = 1, 2, . . ., n− 1, where d i ∈ (0, 1] are some constants. with assumption d 0 = 1.Combining ( 9) and ( 10) The expression given in ( 11) is a general representation for h-order of U n .Further to simplify representation for (11), we consider the following cases.
Case 1.If we can arrange c i and Case 2. If we can arrange c i and . ., X n are iid rvs and Y 1 , Y 2 , . . ., Y n are also iid rvs.Then for Case 1, we have and for Case 2, which are very easy to calculate.Remark 3.3.Observe that, from Theorem 2.1, Theorem 2.2 and Corollary 2.1, if the natural scale function of X i and Y i are h X i and h Y i respectively, then the scale function for U n is given by Next we consider the tail behavior of V n , defined as Now let the constants between (Y i )(max{0, X i + V i−1 }) and (X j ) + (V j−1 ) are c i and d i respectively, where c i ∈ (0, 1] and d j ∈ [ 1  2 , 1] for i = 1, 2, . . ., n and j = 2, 3, . . ., n.Hence, We assume in above expression that d n+1 = 1 = d 1 .Now take Hence, we get )} , which is easy to calculate.We know that V n = Y n max {0, X n + V n−1 } .If the scale function of X i and Y i are h X i and h Y i respectively.Then, using Theorem 2.1, Theorem 2.2 and Corollary 2.1, the scale function h for V n is given by Remark 3.5.As mentioned earlier, U n and V n have same distribution and it can be seen from ( 11) and ( 13), the representation of V n is preferable over the representation of U n due to the ease of computation and simplicity in the construction of h.

Proofs
Proof of Theorem 2.1.Since X 1 , . . ., X n are nonnegative rvs.Using Lemma 1.1, there exist monotone concave functions h 1 , . . ., h n such that E e h i (X) < ∞ and h i (x) = o(x) as x → ∞, for i = 1, . . ., n. Define therefore, h is a monotone concave function and h(0) = 0. Hence, by Remark 2 of [6], h is continuous and increasing function with h(x 1 + x 2 ) ≤ h(x 1 ) + h(x 2 ) for all x 1 , x 2 > 0. We know that max 1≤i≤n x i ≥ x i for i = 1, . . ., n, this implies for s > 0, we have E e sh(max 1≤i≤n X i) ≥ E e sh(X i ) .
Therefore, I h (max 1≤i≤n X i ) ≤ I h (X i ).Hence, Similarly, it is easy to see that Hence combine ( 14) and (15), we get required result.
Proof of Theorem 2.2.Using Lemma 1.1, there exist monotone concave functions h 1 , . . ., h n such that E e h i (X) < ∞ and h Therefore, E e sh( n i=1 X i) ≤ E e s ĥ(X i ) . Hence, This proves (a).It is easy to see that I ĥ(X i ) ≤ I h (X i ), for i = 1, . . ., n. Choose c ≥ 1 such that cI ĥ(X i ) ≥ I h (X i ), therefore, Hence, there exist m ∈ (0, 1] and r ∈ [1, ∞) such that This proves (b) and (c).
Proof of Corollary 2.1.It follows from the proof of Theorem 2.1, To prove the equality, it is known that h ( n i=1 x i ) ≤ n i=1 h(x i ) and X i are independent rvs.Then E e sh( n i=1 X i) ≤ n i=1 E e sh(X i ) .Therefore, I h ( n i=1 X i ) ≥ min 1≤i≤n {I h (X i )} .Hence, This proves (a).Next, from (14), we know that this implies E e sh(max 1≤i≤n X i ) ≤ E e sh( n i=1 X i) ≤ n i=1 E e sh(X i ) . Therefore, Hence,

This proves (b).
Proof of Theorem 2.3.Recall from Theorem 2.2 and Remarks 2.1, we have I h (XY ) = min(I h (X), I h (Y )).For iid rvs I h (X) = I h (Y ).Hence, I h (XY ) = I h (X).Then we can find the tail behavior of the rvs in the following ways.

Remark 3 . 1 .
Suppose, for any value of b, α and β positive, T (X, Y ) = bX α Y β , where X and Y are any nonnegative rvs.Then using the method given in 2.1, we can find the dominated function for T .We demonstrate this phenomenon through following examples, for various conditions of α and β.First, consider for Pareto distribution with parameters a and k, and the condition α = β in Cobb-Douglas production model.Example 3.1.Let X and Y are iid Pareto distributed rvs with common pdf