Prime numbers and Gaussian random walks

Transcription

1 Prime numbers and Gaussian random walks K. Bruce Erickson Department of Mathematics University of Washington Seattle, WA March 24, 205 Introduction Consider a symmetric aperiodic random walk W = {W r } = (W r, W 2 s, W 3 r ), r = 0,,..., on the integer lattice Z 3 starting at W 0 = 0. This means that the steps or increments {W r+ W r } r 0 are independent with a common distribution that is invariant under reflection through the origin and that the support of this distribution generates the entire lattice Z 3. We shall also assume that its one-step displacement has a finite second moment: E W r+ W r 2 < () The symmetry and () implies E (W r+ W r ) = 0 (the vector). (We call describe any walk on Z 3 satisfying all of the preceeding as Gaussian.) The walk induces a non-defective walk on the z-axis. More precisely put T = min{r > 0 : W r = W 2 r = 0} T n = min(r > T n : W r = W 2 r = 0}, n 2, S 0 = 0, and S n = W 3 T(n), n. The two dimensional walk {(W r, W 2 r )} r 0 returns to the origin infinitely often, a.s., so that P{T n < n} =. The Markov property and the linear structure of the z-axis imply that the sequence of increments {X n } {S n S n } is stationary and mutually totally independent. Therefore the sequence S = {S n } n 0, defines a (symmetric)

2 random walk on the one-dimensional lattice Z. (With a slight abuse of terminology, we will call this sequence the walk induced by W on the z-axis.) The well known transience of W implies the transience of S which means that S cannot visit any finite set of integer points more than a finite number of times if at all. Many infinite subsets of lattice points also have this transience property. For example, S will not hit the set of perfect squares, {, 4, 9,... }, infinitely often. Let us denote the set of prime numbers by Q, and by Q the set {0} {0} Q. P. Erdös, H. P. Mckean, C. Stone, and B. Kochen established the interesting result that if W is the simple random walk, then, W returns to the set Q infinitely often. In terms of the induced walk S, P{S n Q i. o.} =. From this result and an important invariance principle due to F. Spitzer, [6], one then gets the recurrence of Q for any aperiodic walk in Z 3 with mean 0 displacement and finite second moments (wether or not the step distribution is symmetric). Although these Gaussian walks W do hit Q infinitely often, it comes as a surprise that for some walks the set Q is transient for the sequence of successive high points on the z-axis. More precisely, let M n = max{s j : j n}, then we prove,under various additional assumptions, that P{M n Q i.o.} = 0. For these random walks, with probability, there is a finite time such that the knowledge that W has reached a new prime point on the z-axis after that time implies that S has already hit a (non-prime) point higher up. (For the exact statement, see Theorem in section 4.) One can certainly conjecture that this assertion remains valid for all Gaussian walks. 2 Characteristic function of the induced walk Let {p n } denote the common distribution of the steps of S and let ϕ(t) denote the characteristic function of X = S, i.e., ϕ(t) = E e its = n p n e int. Our first item is the asymptotic formula ϕ(t) log(/ t ), t 0, 2 (2) The common distribution of the X k is very heavy tailed ; E X q = for any q > 0. 2

3 I do not have a complete proof of the transience of Q for the z-axis ladders for general Gaussian walks. In order to make progress I need to impose some additional restrictions. The first one is not as serious as the second. (See the statement of Theorem below.) Our first additional condition is to assume that the two processes {(W n, W 2 n)} r 0 and {W 3 s } s 0 are independent. (3) Let γ(t) = E exp ( itw 3 ). Then ϕ(t) = E e its = j P{T = j}e e itw3 j = E γ(t) T (4) because T and W 3 are independent by (3). However, it is well known that P{T > n} log n, n (5) (For the simple walk case, see [6], page 67. This result does not depend on the asumption (3). A proof will be given in Appendix 2.) The formula (5) and an Abelian theorem, [2], page 447, ρ = in (5.24)/(5.26), imply that By Taylor s formula E s T s ( s)log ( ), s. (6) s γ(t) = 2 m 2t 2[ + o() ], for t 0. (7) This and (6) and (4) (and the reality of γ by symmetry) imply that ϕ(t) = ( γ(t) ) E γ(t) T γ(t), t 0. log(/ t ) This completes the proof of (2) in this case. 2 We use the notation a(t) b(t), t t 0, to mean that the lim t t0 a(t)/b(t) is finite and strictly positive. The notation a(t) b(t), t t 0, means the ratio of a(t) to b(t) is bounded strictly away from 0 and in a neighborhood of t 0. 3

4 3 Renewal function for the ladders Let N, N, S N, and S N denote the first strict ascending/descending ladder epochs and positions. For example, N + = min{n : S n > 0}. Then, see [2], Chapter XVIII, sϕ(t) = c(s) [ E {s N e its N } ][ E {s N e its N } ] (8) where log c(s) = (s n /n)p(s n = 0). But S N has the same distribution as S N so that on setting s = we find that ϕ(t) = c() χ(t) 2 where χ is the characteristic funcion of S N. From (2) it follows that χ(t) log(/ t ) t 0. (9) Now let f(z) = χ( i logz) = f n z n, n= (f n = P{S N = n}), so that χ(t) = f(e it ), and consider the function log {[ f(z)] 2log( } z), for z, z D(z) = z log C2 2 for z =. (0) The distribution {f n } is aperiodic, that is, χ(t) = if and only if t is a multiple of 2π. This implies that f(z) = and z if and only if z =. (Note that the coefficients of f are non-negative and sum to.) The function log( z) z = n=0 z n n +, is also analytic in z < and has positive derivatives at 0 of all orders. It is also continuous and non-zero at the boundary z = except of course at the point z =. Finally, it never vanishes in z. It follows that D is 4

5 analytic in z <, is continuous on z, and is real valued for z real and positive. Moreover R{D(z)} = log z) [ f(z)]2log( z () Hence D(z) = 2π π π e it + z ( e it z log χ(t) 2 log( e it ) e it ) dt, z <. (2) It follows from this formula, an integral splitting argument, and the asymptotic formula (e it log( e it ) log(/ t ) for t 0, that and consequently lim[ f(s)] 2 log( s) = lime D(s) = C2 2 (3) s s f(s) s C 2 ( s){log( s) } /2, s. (4) Karmata s Tauberian theorem, [2], equation (5.26) on page 447 (with ρ = ), shows that (4) implies q n = coefficient on s n in LHS (4) = k>n f k = P{L = S N > n} log n, n 3 (5) Now consider the sequence {N k } of strict ascending ladder times of the walk S defined inductively by N k+ = min{n > N k : S n > S Nk } (6) This coincides with the sequence of times of occurrence of new maximums of S. Anway if we put L 0 = 0 and L k = S N(k), we get the strict ascending ladder process. Its increments {L k L k } k are mutually independent and have the same distribution {f n } defined earlier. Let ψ n = P{L j = n} = P [ n {L 0, L,..., } ] (7) j=0 3 I want to thank Professor Don Marshall for suggesting the possibility of going from the formula (9) to the formula (3) via the integral (2). 5

6 This is the renewal function for {L k } and its generating function satisfies: ψ n s n = n 0 f(s) Therefore, by (5) and Karamata Tauberians, (8) ψ 0 + ψ + + ψ n log n (9) Conjecture For the sequence {ψ n } as defined above in connection with a given Gaussian walk, we have ( ) ψ n = O n (20) log n Unfortunately, neither (5) nor (9) is sufficient to establish (20). On the other hand the renewal sequence {ψ n } has additional useful properties. In particular let g(w) be the expected number of hits of W at the point w = (w, w 2, w 3 ) and let v n be the expected number of hits by S at n. Then v n = g(0, 0, n). But for general aperiodic random walks on Z 3 with mean 0 and finite second moments we know that g(w) / w for large w. See [6], P, page 308. Therefore v n n, n (2) A sequence {ψ n } is asymptotically decreasing if there are finite positive constants C, n 0 such that ψ m Cψ n for all n and m with m > n n 0 (22) (A bounded increasing sequence is asymptotically decreasing by this definition!) Lemma. If the renewal sequence {ψ n } is asymptotically decreasing, then (20) holds. Proof. For n 0 we will prove in an appendix that for some number c > 0, v n = c ψ k+n ψ k. (23) k=0 6

7 It follows from (23) and (2) and (9) that for some constants C, C 2, etc., and all k sufficiently large, we have C k v k C 2 C 3 ψ 2k k ψ j+k ψ j j=0 k ψ j C 4 ψ 2k log k j=0 And this imples (20) for even n because ρ n = n log n is a regularly varying sequence (with exponent ) so that ρ 2k 2ρ k as k. Applying (22) again gives ψ 2k+ Cψ 2k = O(k log k) ). Thus (20) holds for odd n also. Conjecture 2 The sequence {ψ n } as defined above in connection with a given Gaussian walk is asymptotically decreasing. I have no example for which I can actually verify this. 4 Transience of the primes for the maxima Clearly the random point sets determined by the sequences {L k } and {M n } coincide; the transience of the primes for {M n } is equivalent to the transience of the primes for the ascending ladder height process {L n }. Theorem. Let W = {W r } be a Gaussian walk which satisfies (3) and for which the renewal function for the ladder process {L k } of the induced walk {S n } satisfies (20). If {M n } denotes the sequence of succesive maximum values of {S n }, then P{M n is prime i. o. } = 0. The proof is almost trivial. Let x k = x(k) denote the k-th prime. Then x k k log k as k. Hence if (20) is correct, then ( ) ψ x(k) = O k log k k log(k log k) k=2 k=2 ( ) = O < (24) k (log k) 3/2 k=2 7

8 Clearly (24) is more than enough to imply P{L n is prime i. o.} = 0 which, as noted above, is equivalent to the assertion of the theorem. 5 Appendix : Proof of (23) Write χ(s, t) = E ( ) s N e its N χ (s, t) = E ( s N e its N ) where N is the epoch of the first strict descending ladder height (i.e., the first entrance time to (, 0) of {S n }, when S 0 = 0). Then because S is symmetric, (N, S N ) has the same distribution as (N, S N ), so that χ (s, t) = χ(s, t) = χ(s, t). The Wiener-Hopf facorization (8) then gives sϕ(t) = c(s) [ χ(s, t)][ χ (s, t)] = c(s) χ(s, t) 2 (25) where c(s) = exp[ n (sn /n)p{s n = 0}]. See [2] page 605, formula (3.7). All terms in (25) are continuous 2π-periodic functions of t for s <. Denote the Fourier coefficients of { sϕ(t)} by v n (s) and those of χ(s, t) and χ (s, t) by ψ (s) k and ψ (s) k, and by N j the j-th strict ascending ladder epoch with N 0 = S 0 = 0. Recall that {(N j N j, L j L j )} j= are i.i.d. random vectors, where L j = S N(j). v (s) n = ψ (s) k = ψ (s), k s k P{S k = n}, n = 0 ± ± 2,... k=0 { 0 if k 0 j=0 E { s N j ; L j = k } for k > 0 = ψ (s) k, for all k. From this we see that all of these coefficients are non-negative and form absolutely convergent Fourier series (for fixed 0 s < ). Expanding both sides of (25) in Fourier series and equating coeffiecents on like powers of e it 8

9 and using symmetry of v n, we get, for n > 0, v (s) n = v (s) n = c(s) = c(s) k=0 k= ψ (s) k ψ(s) k+n ψ (s) k ψ(s), n k Also from the formulas for the coefficients, it is clear that ψ (s) k ψ k and v n as s. Therefore, from the last formula and ordinary monotone v (s) n convergence for sums, we obtain v n = limv n (s) = limc(s) s s = c ψ k ψ n+k k=0 k=0 ψ (s) k ψ(s) n+k where c = c() = exp( n (/n)p{s n = 0}) which is non-zero and finite. This completes the proof of (23). The formula (23) does not appear explicitly in either [2] or [6], nor in one other book I consulted. But it must be well known to workers in the field considering the elementary nature of the proof. 6 Appendix 2: Proof of (5) Let h = min[r : P{W r = W 2 r = 0} > 0]. Then for each k the values of T k (the epoch of the k-th return to the z-axis of W) must be integer multiples of h. Moreover the sub-walk { ( W rh, W 2 rh) }r 0 is strongly aperiodic on Z 2 and has mean 0 and finite second moments. See [6], P, page 42. From P9 page 75 of [6] it follows that u n P{T k = nh for some k } = P{W nh = W 2 nh = 0} n so n=0 u ns n log( s) as s. On the other hand u n s n = n=0 E s T /h 9

10 Hence P{T /h > n} = E st /h s n=0 ( s)log( s) as s Karamata s Tauberian theorem now yields P{T > nh} (log n) which certainly implies (5). See [2] equation (5.26), page 447. References [] Benjamini, I., Pemantle, R., and Peres, Y. (995) Martin capacity for markov chains. Annals of Probab [2] Feller, William. (97). An Introduction to Probability Theory and Its Applications, II, 2nd ed. John Wiley & Sons, New York. [3] Kochen, S. and Stone, C. (964). A note on the Borel-Cantelli lemma. Illinois J. Math [4] McKean, H. P. Jr. (96). A problem about prime numbers and the random walk I. Illinois J. Math [5] Ruzsa, I. Z. and Székely, G. J. (982). Intersections of traces of random walks with fixed sets. Annals of Probab., [6] Spitzer, Frank. (976). Principles of Random Walk, 2nd ed. Graduate Texts in Mathematics 34. Springer-Verlag, New York. 0