ON THE TAIL BEHAVIOR OF FUNCTIONS OF RANDOM VARIABLES. A.N. Kumar 1, N.S. Upadhye 2. Indian Institute of Technology Madras Chennai, , INDIA

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1 International Journal of Pure and Applied Mathematics Volume 108 No , ISSN: (printed version; ISSN: (on-line version url: doi: /ijpamv108i112 PAijpameu ON THE TAIL BEHAVIOR OF FUNCTIONS OF RANDOM VARIABLES AN Kumar 1, NS Upadhye 2 1,2 Department of Mathematics Indian Institute of Technology Madras Chennai, , INDIA Abstract: 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 Key Words: Natural scale function, Tail behavior, Ruin probability 1 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 Received: January 21, 2016 Published: June 3, 2016 Correspondence author c 2016 Academic Publications, Ltd url: wwwacadpubleu

2 124 AN Kumar, NS Upadhye We next define the necessary terminology required for the discussion Let X be the rv with cumulative distribution function (cdf F X (x Then F X (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 I(X = liminf x E(X = liminf x R X (x ln(x = sup s 0 : E [( X + s ] <, R X (x = sup s 0 : E ( e sx <, x 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 21 of [6] as follows (X = liminf x R X (x h(x ( = sup s 0 : E e sh(x <, (1 is called the h-order of X and h is said to be natural scale function if (X = 1 Using this definition, the tail of the rv X can be compared as, for ǫ > 0, F X (x e (1 ǫh(x, (2 and for the rvs X and Y, if liminf 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 x N such that for all x > x N, F X (x [ FY (x ] c ǫ (3 Next, we introduce a result of the existence of h satisfying the required properties for discussing the tail behavior of a nonnegative rv Lemma 11 Let X be nonnegative rv Then there exists a monotone concave function h : [0, [0, satisfying h(x = o(x as x and E ( e h(x < Proof It is clear that X is either heavy-tailed or light-tailed rv For heavytailed rv X, Theorem 29 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

3 ON THE TAIL BEHAVIOR OF 125 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 2 The Moment Index for Sum, Maximum and Product of Random Variables Inthissection, weobtain someoftheresultsaboutthetail behavior offunctions of rvs based on the h-order defined in (1 We exemplify the approach of Lemma 11 to various functions of rvs such as sum, maximum and product in the following results Theorem 21 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 ( n ( X i = c 1 min (X i max X i = c 2 min (X i 1 i n 1 i n 1 i n Theorem 22 Let X 1,,X n be nonnegative rvs Then there exist functions h and ĥ such that (a min 1 i n (b ( n (c ( n Iĥ(X i ( n X i min 1 i n (X i X i = m min 1 i n (X i, for some m (0,1] X i = r min Iĥ(X i for some r [1, 1 i n Corollaries 21 Let X 1,,X n be independent nonnegative rvs Then there exists a monotone concave function h such that (a ( n X i = min 1 i n (X i

4 126 AN Kumar, NS Upadhye (b (max X i = min (X i 1 i n 1 i n Remarks 21 1 Observethat, if hsatisfies h( n x i n h(x i, the condition of concavity can also be relaxed (see Theorem 4 of [6] 2 If h satisfies h( n x i n h(x i, then we have ( n X i = min 1 i n (X i 3 Observe that, for X = 0 and h(0 = 0, applying (1, we get (0 = sups 0 : E ( e sh(0 < = sups 0 : E(1 < = Also, if for s > 0, either E ( e sh(x 1 ( = ie, (X 1 = 0 or E ( e sh(x 2 = ( ie, (X 2 = 0 Then (X 1 +X 2 = 0 4 Observe also that, if h(x = x or h(x = ln(x (although ln(x is not a scale function, as h : [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 21 Natural Scale Function Recall from (1 that, a scale function h is called natural scale function for a rv X if (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 f X1,,X n (x 1,,x n dx 1 dx n, R 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 1 = g(x 1,,X n, Y 2 = X 2,,Y n = X n is given by f Y1,,Y n (y 1,,y n = J f X1,,X n (w(y 1,,y n,y 2,,y n,

5 ON THE TAIL BEHAVIOR OF 127 where J is the determinant of the Jacobian matrix J Also, the marginal distribution of Y 1 is f Y1 (y 1 = J f X1,,X n (w(y 1,,y n,y 2,,y n dy 2 dy n ( with supp T and the tail function is given by F Y1 (y 1 = y 1 f Y1 (tdt Hence, the natural scale function is h(y 1 = ln ( FY1 (y 1 We can compare the tail of the random variable Y 1 = g(x 1,,X n with the help of natural scale function h That is, FY1 (y 1 = F (g(x 1,,x n e (1 ǫh(y 1 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, F Y1 (y 1 (the tail function of g is dominated by e (1 ǫh(y 1 Next, we present the upper and lower bounds for the product of two iid rvs Theorem 23 Let X and Y be nonnegative iid rvs Then, for small ǫ > 0, we have the following inequalities, for x > 1, (a If h( = 2R X (, then F XY (x, F X (x e (1 2 ǫh(x (b If h( = R XY (, then F XY (x, F X (x e (1 ǫh(x (c [ FX (x ] 1/(1 ǫ FXY (x [ FX (x ] 1 ǫ Remark 21 In the proof of Theorem 23, 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 3 Applications In this section, we give applications of our results to some well-known models in economics and risk theory 31 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 using

6 128 AN Kumar, NS Upadhye 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 Case (i α + β < 1, there are decreasing return to scale (ie, output decreases proportional to change in inputs Case (ii α + β = 1, there are constant return to scale (ie, output is proportional to change in inputs Case (iii α + β > 1, there are increasing return to scale (ie, output increases proportional to change in inputs Remark 31 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 21, 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 31 Let X and Y are iid Pareto distributed rvs with common pdf f X (x = aka x a+1 where k x < and a,k > 0 Now using technique given in 21 with y 1 = (u /v (u /αv 1/α α/β (1/α 1 β/α g(x,y = x α y β andw(u,v = Hence, it can be easily seen that f(u,v = Therefore, J = (a 2 k 2a /αu (a/α+1 v 1+a (βa/α

7 ON THE TAIL BEHAVIOR OF 129 Since k x,y < implies that k α+β k α v β u ( < Therefore, the marginal distribution of g(x,y is f g(x,y (u = aka (β α 1 /k aα/β u (a/β+1 ( 1 /u (a/α+1 k aβ/α, where k α+β u < The tail function of g(x,y is F ( ( g(x,y (u = ka (β α β /k aα/β u a/β α /u a/α k aβ/α Let ( c = k a/ (β α, then ( / F g(x,y (u = c β k aα/β u a/β ( α /u a/α k aβ/α [ ( and the natural scale function of g(x,y is h(u = ln c β /k aα/β u a/β ( ] α /u a/α k aβ/α From (2, for ǫ > 0 ( F g(x,y (u e (1 ǫh(u = c 1 ǫ β k aα/β u a/β α 1 ǫ u a/α k aβ/α 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 23 for the Pareto distribution Example 32 Suppose X and Y are iid Pareto distributed rv with pdf given by f(x = aka x a+1 where a > 0 and k x < Now using technique given in 21 with y 1 = g(x,y = xy and w(u,v = u/v Therefore, J = 1/v Hence, it is easy to see that f(u,v = ( a 2 k 2a /u a+1 v Since k x < ( implies that k 2 kv u < Also, the marginal pdf of XY is f XY (u = a 2 k 2a ln(u/k 2 /u a+1, wherek 2 u < Thetail function XY is F XY (u = (k 2a( 1+aln(u/k 2 / u a Hence, thenaturalscalefunctionofxy ish XY (u = aln(u ln ( 1+aln ( u/k 2 2aln(k It is clear that XY (X = liminf x R X (x h XY (x = liminf x aln(x aln(k aln(x ln(1+aln( x k 2 2aln(k = 1

8 130 AN Kumar, NS Upadhye Similarly, XY (Y = 1, hence for c = 1, it is satisfied Theorem 21 Compare with production function of Cobb-Douglas model, we have α = β = b = 1 That is, T(X,Y = XY, ie, for x > 1, we have the case in which increasing return to scale, From Theorem 23, the tail function of XY and X are dominated by the function, ( k 2a (1+aln(x/k 2 exp (1 ǫh XY (x = x a 1 ǫ and [ FX (x ] 1/(1 ǫ FXY (x [ FX (x ] 1 ǫ 32 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 W n = n i X i Y j j=1 The basic assumptions for this model (see [9] are as follows A 1 Let A n be the net income within year n Assume A n : n = 1,2, constitute a sequence of iid rvs with support (, 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, A 3 Also, assume A n : n = 1,2, and r n : n = 1,2, are mutually independent Define B n = 1 + r n, also known as the inflation coefficient from year n 1 to year n and let Y n = Bn 1 be the discount factor from year n to year, 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 S 0 = x, S n = B n S +A n,

9 ON THE TAIL BEHAVIOR OF 131 where B n = 1+r n, n = 1,2, The recurrence relation can be written as S 0 = x, S n = x n B j + j=1 n n A i j=i+1 B j, n = 1,2,, (4 where n j=n+1 = 1 Next, the time of ruin for the risk model (4 is defined as τ(x = infn = 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 P(τ(x = n = ψ(x,n ψ(x,, n = 1,2, 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 = infn = 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 S n 0 = x, Sn = S n Y j = x j=1 n i X i Y j = x W n j=1 Hence, for each n = 0,1,2,, ψ(x,n = P(U n > x, where U n = max0, max 1 k n W k with U 0 = 0 Now, define V 0 = 0, V n = Y n max0,x n +V, n = 1,2, Then Theorem L 21 of [9], it is clear that U n = Vn 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 ourresults to discussthe tail behavior of U n UsingTheorem 21, Theorem 22 and Corollary 21, there exists a monotone concave function h such that ( = (U n = (max0, max 1 k n W k max 1 k n W k = c n min 1 k n (W k (5 for some constant c n [ 1 n,1] ( i Suppose constant corresponding to k=1 X ( k k j=1 j Y + X i+1 i+1 j=1 Y j is c i, where c i [ 1 2,1] for i = 1,2,,n 1 Now use Theorem 21 in

10 132 AN Kumar, NS Upadhye W 2,W 2,,W n, we get (W j = min 0 k j 1 j 1 ( c i i=k X k+1 k+1 Y l l=1 (6 with assumption that c 0 = 1, combining (5 and (6, we get (U n = c n min 0 k i=k c i ( X k+1 k+1 Y l, (7 where 0 < c i 1, for i = 1,2,, Now, suppose constant corresponding to X i ( i j=1 Y j is k i, i = 1,2,,n Then, we can write X j i j=1 Y j Combining (7 and (8, we get (U n = c n = min min k m+1 0 m l=1 k j (X j,k j ( j Y j (8 ( m+1 c i (X m+1, Y l (9 Now, again takeconstantscorrespondingto( i j=1 Y jy i+1 isd i, i = 1,2,,n 1, where d i (0,1] are some constants ( n Y i = min d i (Y m+1, (10 0 m with assumption d 0 = 1 Combining (9 and (10 (U n = c n min min 0 m min 0 l k m+1 c i (X m+1, min l m k m+1 l=1 m c i d j (Y l+1 (11 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 j=l

11 ON THE TAIL BEHAVIOR OF 133 Case 1 If we can arrange c i and d i such that d 1 c i1 for i 1 1,2,,n 1,d 2 c i2 for i 2 i 1,, d c i for i i 1 i 2 i n 2 and k n = min 1 p n k p Then Take (U n = c n C = diag min 0 m k 1 i=0 c i,k 2 i=0 k m+1 c i,k 3 c i (X m+1,k n c i=2 c i,,k n c i=2 d i (Y m+1 D = diag k n c d i,k n c d i,k n c d i,,k n c d X = (X 1,X 2,,X n and Y = (Y 1,Y 2,,Y n Then (U n = c n min (XC, (YD Case 2 If we can arrange c i and d i such that c 1 d i1 for i 1 1,2,,n 1, c 2 d i2 for i 2 i 1,, c d i for i i 1 i 2 i n 2 Then (U n = c n min 0 m k m+1 c i (X m+1,k 2 d 1 c i (Y 1,k m+1 d m c i (Y m+1 Take Then C = diag k 1 c i,k 2 c i,k 3 i=2 c i,,k n c D = diag k 2 d 1 c i,k 2 d 1 c i,k 3 d 2 c i,,k n c d i=2 (U n = c n min (XC, (YD

12 134 AN Kumar, NS Upadhye Remark 32 If X 1,X 2,,X n are iid rvs and Y 1,Y 2,,Y n are also iid rvs Then for Case 1, we have (U n = c n min k n c d i (Y 1,mink 1,k 2 c i (X 1, and for Case 2, (U n = c n min k 2 d 1 c i (Y 1,mink 1,k 2 c i (X 1 which are very easy to calculate Remark 33 Observe that, from Theorem 21, Theorem 22 and Corollary 21, if the natural scale function of X i and Y i are h Xi and h Yi respectively, then the scale function for U n is given by h(x = n ( h Xi (x+(n i+1h Yi (x Next we consider the tail behavior of V n, defined as V n = Y n max0,x n +V Now let the constants between (Y i (max0,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, (V n = minc n (Y n,c n (0,X n +V = minc n (Y n,c n (X n +V = minc n (Y n,c n d n (X n,c n d n (V n = min c i d i (X m, 1 m n n c i d i+1 (Y m (12 We assume in above expression that d n+1 = 1 = d 1 Now take n C = diag c i d i+1,,c n d n

13 ON THE TAIL BEHAVIOR OF 135 and Hence, we get n D = diag c i d i+1,,c n (V n = min (XC, (YD Remark 34 If X 1,X 2,,X n are iid rvs and Y 1,Y 2,,Y n are also iid rvs Then (V n = n c id i+1 min (X 1, (Y 1,whichiseasytocalculate We know that V n = Y n max0,x n +V If the scale function of X i and Y i are h Xi and h Yi respectively Then, using Theorem 21, Theorem 22 and Corollary 21, the scale function h for V n is given by h(x = n (h Xi +h Yi Remark 35 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 4 Proofs Proof of Theorem 21 Since X 1,,X n are nonnegative rvs Using Lemma 11, thereexistmonotoneconcave functionsh 1,,h n suchthate ( e h i(x < and h i (x = o(x as x, for i = 1,,n Define h(x = n h i (x n h i (0, 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, (max 1 i n X i (X i Hence, (max 1 i n X i min 1 i n (X i (13

14 136 AN Kumar, NS Upadhye Similarly, it is easy to see that ( n X i (max X i 1 i n (14 Hence combine (14 and (15, we get required result Proof of Theorem 22 Using Lemma 11, 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 Let h(x = n h i(x n h i(0, therefore, h is monotone concave with h(0 = 0 For i = 1,,n, define ĥ(x i = x h(x i, where x = max 1 i n x i +1 It is clear that x i x n x i, This implies ( n ( h(x i h x i /x 1 n x h x i That is, Therefore, ( n h x i x h(x i = ĥ(x i E (e sh( n X i ( E e sĥ(x i Hence, Now, consider E (e sh( n X i ( n X i min Iĥ(X i 1 i n E (e sh( n X i 1(X2 1,,X n 1 E (e n sh(x 1 P(X i 1, we get ( n X i (X 1, similarly, for i = 2,,n, ( n X i (X i Hence, ( n X i min 1 i n (X i Hence, min 1 i n Iĥ(X i ( n X i min 1 i n (X i i=2

15 ON THE TAIL BEHAVIOR OF 137 This proves (a It is easy to see that Iĥ(X i (X i, for i = 1,,n Choose c 1 such that ciĥ(x i (X i, therefore, 1 c min I h(x i min Iĥ(X i ( n X i 1 i n 1 i n min 1 i n (X i c min 1 i n Iĥ(X i Hence, there exist m (0,1] and r [1, such that ( n ( n X i = m min I h(x i and X i = r min Iĥ(X i 1 i n 1 i n This proves (b and (c Proof of Corollary 21 It follows from the proof of Theorem 21, ( n X i min 1 i n (X i To prove the equality, it is known that h( n x i n h(x i and X i are independent rvs Then E (e sh( n X i n E( e sh(x i Therefore, ( n X i min 1 i n (X i Hence, ( n This proves (a Next, from (14, we know that (max X i 1 i n Now, it is clear that this implies X i = min 1 i n (X i max x i 1 i n min 1 i n (X i n x i, ( E e sh(max 1 i nx i E (e sh( n X i n ( E e sh(x i

16 138 AN Kumar, NS Upadhye Therefore, Hence, This proves (b (max X i min (X i 1 i n 1 i n (max X i = min (X i 1 i n 1 i n Proof of Theorem 23 Recall from Theorem 22 and Remarks 21, we have (XY = min( (X, (Y For iid rvs (X = (Y Hence, (XY = (X Then we can find the tail behavior of the rvs in the following ways Case 1 Suppose h 1 is a natural scale function of X, therefore h = 2h 1 and (X = (X = 1/2 Then (XY = (X = 1/2, for small ǫ > 0, using the definition of lim inf, F XY (x, F X (x e (1 2 ǫh(x This proves (a Since, h 1 isthenaturalscalefunctionx, therefore, h(x = 2h 1 (x = 2R X (x and (XY = 1/2, hence, that is, From (3, lim inf x lim inf x R XY (x 2R X (x = 1/2, R XY (x R X (x = 1 F XY (x [ FX (x ] 1 ǫ (15 Case 2 Using the method described in 21, we can find the natural scale function h for XY then (XY = (X = 1 Using the definition of liminf, for ǫ > 0, F XY (x, F X (x e (1 ǫh(x This proves (b Next, note that h is the natural scale function XY, therefore, h(x = R XY (x and (XY = (X = 1, that is, (X = 1, hence, lim inf x R X (x R XY (x = 1

17 ON THE TAIL BEHAVIOR OF 139 Therefore, from (3, F X (x [ FXY (x ] 1 ǫ (16 Combining (16 and (17, it is clear that [ FX (x ] 1/(1 ǫ FXY (x [ FX (x ] 1 ǫ This proves (c References [1] AA Balkema, L DeHaan, Residual Lifetime at Great Age, Ann Prob, 2 (1974, [2] C Yu, Y Wang, D Cheng, Tail behavior of the sums of dependent and heavy-tailed random variables, To appear in Journal of the Korean Statistical Society, (2014 [3] C W Cobb, P H Douglas, A Theory of Production, American Economic Review, 18 (1928, [4] D J Daley, The moment index of minima, J Appl Prob, 38 (2001, [5] D J Daley, C M Goldie, The moment index of minima (II, Statistics & Probability Letters, 76 (2006, [6] J Lehtomaa, A comparison method for heavy-tailed random variables, Preprint arxiv: , (2013 [7] M C Bhattachaijee, Tail Behavior of Age-Smooth Failure Distributions and Applications, Reliability and Quality Control, ed A PBasu, Amsterdam, North-Holland (1986 [8] P C H Phillips, A Theorem on the Tail Behaviour of Probability Distributions with an Application to the Stable Family, The Canadian Journal of Economics, 18 (1985, [9] Q Tang, G Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risk, toch Proc Appl, 108 (2003, [10] R V Hogg, J W McKean, A T Craig, Introduction to mathematical statistics (sixth edition, Prentice Hall, Upper Saddle River, New Jersey (2005 [11] S Foss, D Korshunov, S Zachary, An introduction to heavy-tailed and subexponential distributions, Springer Verlag, New York (2011

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