On asymptotic behaviour of the increments of sums of i.i.d. random - PowerPoint PPT Presentation

On asymptotic behaviour of the increments of sums of i.i.d. random variables from domains of attraction of asymmetric stable laws. Terterov M., St-Petersburg State University March, 2010 1 / 17

Let X, X 1 , X 2 , . . . be a sequence of independent identically distributed (i.i.d.) random variables. Put S n = X 1 + . . . + X n , S 0 = 0 . Let a n be a nondecreasing sequence of natural numbers. We will study the asymptotic behaviour of the increments of sums T n = S n + ca n − S n as well as the maximal increments U n = 0 ≤ k ≤ n − a n ( S k + a n − S k ) . max The aim is to describe a normalizing sequence c n such that lim sup T n = 1 a.s. c n lim sup U n = 1 a.s. 2 / 17 c n

L. Shepp (1964) T n = S n + an − S n , a n ր ∞ , a n takes positive a n integer values. M t = E e Xt < ∞ . T = lim sup T n was determined in terms of the moment generating function of X and the radius of convergence of � x a n (denoted r ). m ( a ) = min M ( t ) e − at . T = a a.s., where a = a ( r ) is the unique solution of m ( a ) = r . 3 / 17

P. Erd˝ os, A. R´ enyi (1970) a n = [ c log n ] . Theorem 1. Suppose that the moment generating function M t = E e Xt exists for t ∈ I , where I is an open interval containing t = 0 . Let us suppose that E X = 0 . Let α be any positive number such that the function M ( t ) e − αt takes on its minimum in some point in the open interval I and let us put t ∈ I M ( t ) e − αt = M ( τ ) e − ατ = e − 1 /c . min Then S k +[ c log n ] − S k P (lim max = α ) = 1 [ c log n ] 0 ≤ k ≤ n − [ c log n ] 4 / 17

Theorem 2. The functional dependence between α and c = c ( α ) determines the distribution of the random variables X n uniquely. Practical implements. 1. The longest runs of pure heads. Theorem 3(special case of Theorem 1). Let X 1 , X 2 , ... be independent Bernoulli random variables with P ( X i = 1) = 0 . 5 = P ( X i = − 1) , S n = X 1 + . . . + X n . Then for any c ∈ (0 , 1) there exists n 0 = n 0 ( c ) such that 0 ≤ k ≤ [ c log 2 n ] ( S k +[ c log 2 n ] − S k ) = [ c log 2 n ] max a.s. if n > n 0 . This theorem guarantees the existence of a run of length [ c log 2 n ] when n is large enough. 5 / 17

2. The stochastic geyser problem. X 1 , X 2 , ... - i.i.d.r.v., F ( . ) is their distribution function. Put V n = S n + R n , where R n is also a r.v. sequence. Theorem (B´ artfai, 1966). Assume that the moment generating function of X 1 exists in a neibourhood of t = 0 and R n = o (log n ) . Then, given the values of { V n ; n = 1 , 2 ... } , the distribution function F ( . ) is determined with probability 1, i.e. there exists a r. v. L ( x ) = L ( V 1 , V 2 , ..., x ) , measurable with a respect of σ -algebra, generated by V 1 , V 2 ... such that for any given real x , L ( x ) = F ( x ) . Proof. For any c > 0 we have V k +[ c log n ] − V k lim max = [ c log n ] 0 ≤ k ≤ n − [ c log n ] S k +[ c log n ] − S k lim max = α ( c ) a.s. [ c log n ] 0 ≤ k ≤ n − [ c log n ] 6 / 17

Improvements. J. Steinebach (1978). The existence of a moment generating function is a necessary condition. If M ( t ) = E e Xt = ∞ for all t > 0 , then S k +[ c log n ] − S k lim sup max = ∞ a.s. [ c log n ] 0 ≤ k ≤ n − [ c log n ] D. Mason.(1989) (The extended version of Erd˝ os-R´ enyi laws). S k + a n − S k a.s. max → 1 , γ ( c ) a n 0 ≤ k ≤ n − a n where γ ( c ) is a constant depending on c and M ( t ) remains true when a n / log n → 0 . (Erd˝ os and R´ enyi had a n / log n ∼ c ) . 7 / 17

M. Cs¨ org˝ o and J. Steinebach (1981). Theorem. Suppose E X = 0 , E X 2 = 1 and there exists a t 0 > 0 such that M ( t ) = E e Xt < ∞ if | t | < t 0 . Then for the sums S n the following holds S k + a n − S k lim max (2 a n log( n/a n )) 1 / 2 = 1 a.s., n →∞ 0 ≤ k ≤ n − a n a n where (log n ) 2 → ∞ . In this case the normalizing sequence depends only on the moment conditions on X . 8 / 17

T n = S n + ca n − S n lim sup U n U n = 0 ≤ k ≤ n − a n ( S k + a n − S k ) , max = 1 a.s. c n The asymptotic behahior of U n and T n strongly depends on the rate of the growth of a n and the moment conditions on X . If a n = O (log n ) , the normalizing sequence c n depends on the distribution of X (Erd˝ os-R´ enyi laws). If a n / log n → ∞ and E X = 0 , E X 2 = 1 , the normalising sequence does not depend on the distribution of X and is the same as the one for the Gaussian distribution. In this case � c n = 2 a n (log( n/a n ) + log log n ) (Cs¨ org˝ o-R´ ev´ esz laws). For example: put a n = n , c n = (2 n log log n ) 1 / 2 , U n = S n , S n lim sup √ 2 n log log n = 1 a.s. 9 / 17

Frolov (2000). It turned out, that these two types of behaviour are particular cases of the universal one. For variables with a finite moment generating function there exists an explicit formula for the normalizing sequence c n . 10 / 17

H. Lanzinger, U. Stadtmuller. Let X, X 1 , X 2 , . . . be a sequence of i.i.d. random variables. Suppose E X = 0 , E X 2 = σ 2 . E e t | X | 1 /p < ∞ for all t in a neibourhood of 0. t 0 = sup { t ≥ 0 : E e g ( tX ) < ∞} ∈ (0 , ∞ ) x 2 1 p ≤ 1 , x ≥ 0 , y ≥ 0 } . ϕ ( c ) = max { x + y : 2 cσ 2 + ( t 0 y ) Theorem. Under assumptions made above, we have S j + k − S j n →∞ max lim max (log n ) 2 p − 1 )(log n ) p = 1 a.s. k ϕ ( 0 ≤ j<n 1 ≤ k ≤ n − j 11 / 17

Corollary S j + c (log n ) 2 p − 1 − S j lim sup max = 1 a.s. ϕ ( c )(log n ) p 0 ≤ j<n n →∞ H. Lanzinger (2000). Theorem. S n +(log n ) p − S n lim sup = ϕ (1) a.s. (log n ) ( p +1) / 2 n →∞ 12 / 17

Definition. Suppose that X has a distribution R . The distribution R is stable if for every n there exist c n > 0 and γ n such that S n = c n X + γ n . c n = n 1 /α c , 0 < α ≤ 2 . Normal distribution is stable with α = 2 and γ n = 0 . The distribution function G belongs to the domain of attraction of R if there exist a sequence B n , B n > 0 and A n , such that S n − A n d → R. B n 13 / 17

There exists a canonical representation of the characteristic function of a stable law. f ( t ) = exp( itγ − c | t | α (1 − i t | t | βω ( t, α ))) , where γ ∈ R , c ≥ 0 , | β | ≤ 1 , ω ( t, α ) = tan πα/ 2 if α � = 1 and ω ( t, α ) = (2 /π ) log t , if α = 1 . 14 / 17

Let X, X 1 , X 2 , . . . be a sequence of i.i.d. random variables, E X = 0 , F ( x ) = P ( X < x ) . Suppose F ( x ) to be from a domain of attraction of a stable law with index α ∈ (1 , 2) and the characteristic function ψ ( t ) = exp {− a | t | α (1 + i t | t | tan π 2 α ) } , 1 α . a = cos( π (2 − α ) / 2) . Let B n = n Define, further α p + α − 1 < ∞} , p + α − 1 t 0 = sup { t ≥ 0 : E e t ( X + ) c n = (log n ) , α α ϕ ( c ) = max { x + y : ( α − 1) x α − 1 α p + α − 1 ≤ 1 , x ≥ 0 , y ≥ 0 } . + t 0 y 1 αc α − 1 Theorem. Suppose t 0 ∈ (0 , ∞ ) Then S n + ca n − S n lim sup = 1 a.s. c n ϕ ( c ) n →∞ 15 / 17

References. Cs¨ org˝ o M., Steinebach J., Improved Erdos-Renyi and strong � approximation laws for increments of partial sums, Ann. Probab. v. 9, 1981, 988-996. Erd˝ os P., R´ enyi A., On a new law of large numbers, J. � Analyse Math., v. 23, 1970, 103-111. Frolov A. N., One-sided strong laws for increments of sums � of i.i.d. random variables, Studia Scientiarum Mathematicarum Hungarica v. 39, 2002, 333-359. Lanzinger H., A law of the single logarithm for moving � averages of random variebles under exponential moment condition, Studia Scientiarum Mathematicarum Hungarica, v. 36, 2000, 65-91. 16 / 17

Lanzinger H., Stadtmuller U., Maxima of increments of � partial sums for certain subexponential distributions, Stochastic Processes and their Applications, v. 86, 2000, 307-322. D. Mason, An Extended Version of the Erdos-Renyi Strong � Law of Large Numbers, The Ann. Probab., Vol. 17, No. 1 (Jan., 1989), pp. 257-265. Shepp L. A., A limit law concerning moving averages, Ann. � Math. Statist., v. 35, 1964, 424-428. J. Steinebach, On a Necessary Condition for the � Erd˝ os-R´ enyi Law of Large Numbers, Proceedings of the AMS, Vol. 68, No. 1 (Jan., 1978), pp. 97-100. 17 / 17