On the asymptotic growth rate of the sum of p-adic-valued independent identically distributed random variables

AbstractThe p-adic semistable laws are characterized as weak limits of scaled summands of p-adic-valued, rotation-symmetric, independent, and identically distributed random variables whose tail probability satisfies some condition. In this article it is verified such scaled sums do not converge in probability, and some more precise estimates, corresponding to the law of iterated logarithm in the real-valued setting, are given to the asymptotic growth rate of the sum. The critical scaling order is explicitly given, over which the scaled sum almost surely converges to 0. On the other hand, under the critical order, the limit superior of the p-adic norm of the scaled sum diverges almost surely. Furthermore it is shown that, at the critical order, a crucial change of the asymptotic behavior of the scaled sum occurs according to the decay of the tail probability of the random variables. In this situation, the critical value for the order of the tail probability is also found.


that P(ξ ∈ X ) = P(ξ ∈ u X) for any measurable subset X of the p-adic field and any unit u in the ring of the p-adic integers. In case the ξ i are bounded and non-degenerate, the ultra-metric property of the p-adic norm brings a different situation from the Euclidean case; the distribution of the sum, without any scaling, converges to the normalized Haar measure on the minimum disc centered at the origin that contains the support of ξ i . On the other hand, if we consider unbounded ξ i , "p-adic semistable laws" appear as the weak limit distributions of the scaled sums. In this article, we will show the convergence to the p-adic semistable laws does not follow in stronger senses, and determine the precise growth rate of the sum.
Throughout this article, we fix a prime integer p. For a non-zero integer z, the p-adic norm of z is defined by |z| p = p −m , where m is the maximum integer such that p m divides z. The p-adic norm is naturally extended to rational numbers, if we define |0| p = 0 and z z p = |z| p |z | p for integers z and z = 0. The field of p-adic numbers, denoted by Q p , is the topological closure of Q with respect to the p-adic norm. The non-zero elements of Q p are identified with formal series x = ∞ i=m a i p i with m ∈ Z, a i ∈ {0, 1, 2, . . . , p − 1}, a m = 0, and they are normed by |x| p = p −m . The p-adic norm satisfies the ultra-metric inequality : x + x p ≤ max |x| p , x p , and this characteristic property is frequently used for some estimations in the followings.
A character on Q p is a continuous homomorphism on the additive group of p-adic numbers to the multiplicative group of complex numbers with absolute value 1. For p-adic numbers x, let us define Then ϕ 1 is a character on Q p , and any character ϕ is of the form ϕ(x) = ϕ y (x) := ϕ 1 (yx) for some y ∈ Q p . For a distribution μ on the field of p-adic numbers, its characteristic function is defined on the group of characters byμ(ϕ) := Q p ϕ(x)μ(dx). The characteristic function is considered to be a function on Q p through the identification ϕ y ↔ y, and written asμ(y) :=μ(ϕ y ), y ∈ Q p . Definition 1.1 For α > 0, a distribution μ on the field of p-adic numbers is called a ( p-adic) α-semistable law, if its characteristic function is given bŷ for some constant c > 0.
For p-adic-valued i.i.d. ξ i (i = 1, 2, . . . ), we shall denote their tail probability by T (s) := P(|ξ i | p ≥ s), s ≥ 0, and let S n := n i=1 ξ i be their partial sums. The p-adic semistable laws are characterized as limit distributions of scaled sums of rotation-symmetric i.i.d. as follows.

Proposition 1.2 ([4] Theorem 2) A distribution μ on the field of p-adic numbers is semistable, if and only if
it is a non-degenerate limit distribution of the scaled sum p n S k(n) for some rotation-symmetric i.i.d. ξ i (i = 1, 2, . . . ) and some increasing sequence {k(n)} n=1,2,... of natural numbers satisfying sup n k(n)T p n+l < +∞ for any integer l.
In what follows, the integer part of a real number t is denoted by [t]. For a random variable X , L(X ) denotes the law of X , andL(X ) its characteristic function. We let μ α be the α-semistable law with the characteristic functionμ α (y) = e −|y| α p . We shall examine next what happens at the critical order c n ∼ log n α log p . As the following theorem shows, the both cases lim sup n→∞ p n+c n S [N (n)] p = 0 and = +∞ may occur according to the order of the convergence L(n+1) L(n) → 1. Proofs to Theorems 1.4 and 1.5 will be given in Sect. 4.

Convergence of the scaled sum
In the followings let α > 0, {L(n)} be a sequence of positive numbers such that lim n→∞ L(n+1) L(n) = 1, and ξ i be p-adic-valued, rotation symmetric i.i.d. with the tail probability T ( p n ) = p −αn L(n). Besides the weak convergence confirmed in Proposition 1.3, let us verify whether the scaled sum p n S [N (n)] converges in any stronger sense.

Proposition 2.1 Under the condition of Proposition 1.3, the scaled sum p n S [N (n)] does not converge almost surely.
Proof We have and by Proposition 1.3 and 1 − p n → 1 in Q p , the law of (1 − p n ) p n S [N (n)] converges to μ α . Since the random variables p 2n are independent, comparing the characteristic functions of the both sides of (1) gives for each p-adic number y.
converges to 0 almost surely. Then the left-hand side of (2) should tend to 1, which is a contradiction.
Furthermore, we can show the convergence in Proposition 1.3 is not even in probability. Proof Take a random variable X α whose law is μ α . If we suppose p n S [N (n)] converges to X α in probability, then for any ε > 0 we have P p n S [N (n)] − X α p > ε → 0 as n → ∞. The ultra-metric property implies converges to 0 in probability, and hence in law. Then we have (2) again.

Limit superior of the norm of the scaled sum
Next let us study the limit superior of p n S [N (n)] p . For an integer m and natural numbers k > l, put since the random variables S k−1 − S l and ξ k are independent. Inductively we can derive that Here let us put N = [N (n)] − [N (n − 1)] and t = 2T p n+s , then Since L(n) L(n−1) → 1, we can take M ≥ 1 such that p −α/2 < L(n) L(n−1) < p α/2 holds for all n ≥ M. Then the above implies Since N ≥ N (n) − N (n − 1) − 1 → +∞ as n → ∞, the right-hand side of (4) tends to 1 2 1 − e −η > 0. Hence we obtain ∞ n=1 R n,s = +∞.
Here note that for t > 0 and a natural number N , and the each term in these sums is estimated as If we take M ≥ M sufficiently large, then for any n ≥ M the last term in (9) is less than 3, and thus the right-hand side of (8) is positive. Hence we can derive from (6) and (7) that provided n ≥ M . Accordingly, the assumption S < +∞ implies Here for 0 < t < 1 and a natural number N , we see and the each term in these sums is estimated as Since L(n+1) L(n) → 1 and T ( p n ) → 0 as n → ∞, there exists M ≥ 1 such that p −α/2 < L(n+1) L(n) < p α/2 and T p n+c n +s < 1 2 hold for all n ≥ M. Put N = [N (n)] − [N (n − 1)] and t = 2T p n+c n +s , then for n ≥ M we have Since L(n) L(n−1) → 1, p −α(c n +s)/2 → 0, and T p n+c n +s → 0 as n → ∞, we can take M > M such that the right-hand side of (12) is less than 1 for any n ≥ M . Then for n ≥ M the right-hand side of (11) is positive, If M is sufficiently large, we can assume N (n) − N (n − 1) ≥ 2, and then (10) leads to × L(n + c n + 1) L(n + c n ) · L(n + c n + 2) L(n + c n + 1) · · · L(n + c n + s) L(n + c n + s − 1) Then the assumption S = +∞ implies If we suppose S [N (n)] (ω) p < p n+c n +s for all large n, then by the ultra-metric property,