The Longest Run of Heads

Transcription

1 The Longest Run of Heads Review by Amarioarei Alexandru This aer is a review of older and recent results concerning the distribution of the longest head run in a coin tossing sequence, roblem that arise in many alications and fields such as quality control, realiability, etc.). First, we will describe exact and recurrent formulas for the distribution of the longest run, both in the case when the coin is fair = q = ) or biassed 0, )). We will observe that for long 2 and very long sequences the exact formulas are not of much hel and the need for bounds and asymtotic analysis is mandatory. The second section concerns with these roblems. Finally we consider that is worth mentioning some extensions to other run related henomena.. The exact and recurrent formulas for the distribution of the longest run of heads We will describe in the following an interesting and simle classroom exeriment, due to T. Varga as mentioned in Révész [2] or Schilling [3], from which the roblem of finding the length of the longest ure heads in n Bernoulli indeendent trials arose naturally. The exeriment begins by dividing a class into two grous. In the first grou, students had to toss a coin 200 times the coin is considered fair) and record the resulting sequence of heads and tails. In the second grou, each student had to write down a random simulation of 200 coin tosses. After collecting the sheets of aer from the two grous, based on a simle argument, the students can be classified into their original grous with a surrising degree of accuracy. The argument is based on the following observation: in students simulated sequences, the longest run of the consecutive heads or tails is too short maximum 5 units long) relative to that which arise from the natural coin tossing rocess which is about 7 H or T long). The exeriment described lead to a natural question: what is the exected value for the length of the longest head run in n tosses of a fair coin? We will try to answer this question and others in subsequent... The case of a fair coin. In 948, Bateman [] gave an exact formula for the distribution of the longest run, in both fair coin or biassed coin cases. He started by considering a sequence of n elements realizations of Bernoulli indeendent trials), r = k of which ossessed a roerty E and r 2 = n k which don t osses that

2 2 roerty are Ē), under the null hyothesis, H 0, that the elements of the sequence are in random order. His idea was to consider the artitions of r and r 2 having s as the greatest art, where s =, 2,..., L and L is the given length for the longest run, and determine the different ways in which such artitions form 2t or 2t + grous, where t =, 2,..., r L + for r r 2. He also exloited the fact that the number of artitions of r i=,2 into t arts of which the greatest art contains s elements can be exressed in terms of binomial coefficients, as this number reresents the coefficient of x r i in the exansion x + x x s ) t = x ) t x s t. x He obtained the following formula: ) PL n m k, n k) = [ m] k Cn k ) j+ C j n k+ Cn k n jm where L n is the length of the longest run of heads in a sequence of n Bernoulli trials coin tosses). He also obtained a formula for the distribution of L n in the biassed case, i.e. the robability of head or success) is 0, ), but this formula will be given in the next section. On the same conditions, and using the observation that s failures divide the sequences of successes into t = s + runs and that the number of ermutations with longest success run < m is equal to the number of ways of selecting in order t integers whose sum is k, each between 0 and m, i.e. the coefficient of x r in the exansion of + x + x x m ) t, Burr and Cane [3] obtained the following j= formula for distribution of the length of the longest run: 2) PL n < m n, k) = [ C n k Cn n k n Cn k+c n m n k + Cn k+c 2 n 2m n k... ] which is related to ). simulation for moderate m): 3) PL n < m n, k) = tkm) n m) They also gave a kind of recurrence relation useful in { t [ k m) m) t 2 ]} k 2m) m)... ) 2 n m) m) 3 n 2m) m) where we used the notation a b) = aa )... a b + ) if a b and a 0) = 0 if a < b. A recurrent formula for the distribution of the longest head run was obtained by Schilling [3] in both, fair or biassed coin, cases. Here we will describe the formula for the first case. If we denote the number of sequences of length n in which the longest run of heads does not exceed m, by A n m), it is easy to see that the robability

3 of length of the longest head run does not exceed m is equal with Anm) 2 n. The key to comute A n m), as Schilling mentioned, is to artition the set of favorable sequences according to the number of heads that occur before the first tail if any). Doing this we obtain that if n m than A n m) = 2 n and if n > m each favorable sequence begins with one of the following subsequences: T, HT,..., HH... HT the last subsequence contain m heads) and is followed by a string having no more than m consecutive heads, which lead to: { m j=0 4) A n m) = A n j m), for n > m 2 n for n m. An interesting fact is that the number of sequences of length n that contain no consecutive heads, A n ), is a Fibonacci number. Using this recursion formula we can easily comute the values for A n 4): n A n 4) so the robability that the length of the longest run is not greater than 4, in n = 0 coin tosses is = The case of a biassed coin. Now we will consider the case in which we have a sequence of n Bernoulli trials with the robability of success or of head) to be = q 0, ). As we mentioned in the revious case, Bateman [] found a formula for the distribution of the length of the longest head run: [ m] n 5) PL n m) = ) j+ + n jm + ) q C j n jm j jm q j j= The equation 5) was deduced from the equation ) by: n 6) PL n m) = PL n m k) Pk) k=m and noticing that Pk) = Cn k k q n k and n k=0 Ck n k q n k =. Using multinomial coefficients, Philiou and Makri [] found exact formulas for the distribution of the number of head runs of length k, denoted by N n k), and for the length of the longest head run L n ) in n indeendent coin tosses Bernoulli trials). They observed, the same argument being found in Schilling [3], that a tyical element N n k) = x) is an arrangement of the form: a a 2... a x +x 2 + +x k +x ss... s }{{} i 3

4 4 for 0 i k and such that x of the a s are T, x 2 of the a s are HT,...,x k of the a s are H T... H }{{} k and x of the a s are H }.{{.. H}. Notice first that between x,...,x k we k have the relation k j= jx j + kx + i = n and that the number of such arrangements is given by the multinomial coefficient: Their results are as follows: k 7) PN n k) = x) = i=0 x,...,x k x + + x k + x x,..., x k, x ) ) ) x + +x x + + x k + x q k n x,..., x k, x where the inner summation is over all nonnegative integers x,..., x k such that the relation k j= jx j + kx + i = n is verified, 8) PL n k, S n = r) = r q n r k i=0 x,...,x k+ ) x + + x k+ x,..., x k+ where S n denote the number of heads in the sequence and the inner summation is over all nonnegative integers x,..., x k such that k+ j= jx j = n i and such that k+ j= x j = n r, k ) x + + x k+ 9) PL n k S n = r) = C r n i=0 x,...,x k+ x,..., x k+ in the same conditions as 8). We will end this section by giving a recurrent formula for the distribution of the length of the longest run of heads, due tu Schilling [3]. Unlike the case of a fair coin 4), in the case of a biassed coin with the robability of heads ) we need to consider, in addition to the length of the sequence, the total number of heads S n = k). Let the number of strings of length n in which exactly k heads occur and such that no more than m of these occur consecutively be denoted by C n k) m). Then the robability that the length of the longest head run does not exceed m can be exressed as: PL n m) = n k=0 C n k) m) k q n k Using the same notation as in the case of a fair coin, we can observe that the number of sequences of length n in which the longest run of heads does not exceed m can

5 be written as: A n m) = n k=0 C n k) m) If we use the same argument of ortioning as in the fair coin case we get the following recursion formula: 0) C n k) m) = m j=0 Ck j) n j m), C m n, for n > k > m for k m 0 for m < k = n The asymtotic behavior of the longest head run We can notice that the exact formulas, and even the recurrent ones, are not of much hel when we deal with long sequences or the ratio n is big). Take for m examle n = 00, 000 tosses and look for the robability of getting a run of at least 25 heads of a fair coin, then the exact formula 5) involves a sum of [ n m] = 4000 terms. For the case in which n is large a series of aroximations, bounds and asymtotic formulas were develoed. 2.. The case of a fair coin. First we will consider the case when the robability of head of success) is. Erdös and Révész [5], using a result of Erdös and Rényi [4], 2 were interested in bounds for the length of the longest head run that would hold for almost all ω Ω where Ω denotes the basic sace) and in some cases they obtained the best ossible results. Let X, X 2,..., X n be a sequence of indeendent and identically distributed Bernoulli random variables such that PX i = ) = PX i = 0) = 2, and let L n be the length of the longest head run among first n coin tosses, i.e. L n is the larger integer L such that there is a k, k n L +, with X k = X k+ = = X k+l =. We will give this results below: Theorem 2. Th. 3 & 4 from [5]). Let α n a sequence of ositive integers. If 2 αn =, n= then for almost all ω Ω there is an infinite sequence n i = n i ω, α n ) n ), i, of integers such that L ni α ni, i.

6 6 If, on the other hand, 2 αn <, n= then for almost all ω Ω there exists a ositive integer n 0 = n 0 ω, α n ) n ) such that L n < α n, n n 0. The following theorem, due to Erdös and Rényi [4], gives bounds for L n : Theorem 2.2. Let 0 < C < < C 2 < then for almost all ω Ω there exists a finite integer N 0 = N 0 ω, C, C 2 ) such that if n N 0. [C log 2 N] L n [C 2 log 2 N] Starting from the Theorem 2.2, Erdös and Révész obtained in [5] sharer bounds for L n. The next theorem illustrates this: Theorem 2.3 Th. & 2 from [5]). Let ɛ be any ositive number. Then for almost all ω Ω there exists a finite integer N 0 = N 0 ω, ɛ) such that for n N 0, L n [log 2 n log 2 log 2 log 2 n + log 2 log 2 e 2 ɛ]. Also for almost all ω Ω there exists an infinite sequence n i = n i ω, ɛ), i, of ositive integers such that L n < [log 2 n i log 2 log 2 log 2 n i + log 2 log 2 e ɛ]. We observe that between the bounds given in the Theorem 2.3 there is a ga which is essentially equal to unity. Guibas and Odlyzko gave in [7] sharer bounds for L n, they also extend and deeen these results to include arbitrary reetitive atterns of heads and tails and that s the reason why we will discuss about their results in the final section. In his article, Turi [4] gave some interesting results regarding the limit behavior of the longest run of ure heads and of ure heads or ure tails. He stated that, under some conditions, the distribution of the number of ure head runs of length k and also for the ure heads or ure tails), N n k), converges to a comound Poisson distribution: Theorem 2.4 Th.2.2 in [4]). If n and k such that n 2 k+ λ

7 7 then we have Es N k) n ) e λ 2) z 2 z ). Theorem 2.5 Th.2.4 in [4]). For any integer k we have PL n [log 2 n] < k) = e 2 k+ {log 2 n}) + o) where [a] denotes the integer art of a and {a} = a [a]. If we ask about the exectation or the variance of the length of the longest head run, Boyd found some interesting results in [2], using generating functions. comuted the exected maximum run of losses in n trials, denoted by Mn), by the formula: ) Mn) = ln n + γ ln ɛ ) ln n + O ln 2 ) ln n) 4 where γ is Euler s constant, ɛ α) is a function of eriod which satisfies for all α: ɛ α) < He also found that the variance V arl n ) is bounded for n and the following relation is valid: 2) V n) = 2 + π2 6ln 2) 2 + ɛ 2 ) ln n + O ln 2 n ) ln n) 5 where ɛ 2 α) is a function of eriod which satisfies for all α: ɛ 2 α) < 0 4. We have to observe, however, that the functions ɛ α), ɛ 2 α) ossesses no limit and that leads to the conclusion that the longest head runs ossesses no limit distribution. Note that from the above formula for the variance we ) can deduce that the standard deviation is aroximately V n)) 2 + π2 2 =.873, a very small value 2 6ln 2) 2 that imlies the fact that the length of the longest head run is quite redictable is within about two of its exectation). In order to rove the results ) and 2), Boyd showed first that the function F n, k), the robability that there is no losing run of length greater than k in a sequence of n trials, has an interesting limiting behavior Th. in [2]). He showed that, as n, uniformly in k: ) 3) F n, k) e 2 k+log 2 n) ln n) 3 = O n The result is very useful and a generalization, due to Gordon, Schilling and Waterman in [6], will be resented in the next section. n He

8 The case of a biassed coin. Now we consider the case in which we have a sequence of Bernoulli trials with robability of head or success) equal to 0, ). Like in the revious section we try to find bounds and the asymtotic behavior for the length of the longest head run. Let use the notation Rk; n, ) = PL n < k) for the robability that the length of the longest head run is strictly less than k. The following theorem, due to Muselli [9], gives an inequality used subsequent in the aer to determine different bounds for Rk; n, ): Theorem 2.6 Th. and Cor. in [9]). If < k < n the following inequalities holds: 4) Rk; m, )Rk; n m + k, ) < Rk; n, ) < Rk; m, )Rk; n m, ) for every integer m such that k m < n and the left inequality being also true for 0 m < k. Noticing that for m = k we have Rk; k, ) = k relacing m = k = n in equation 5)) and using the inequality 4), we obtain k ) Rk; n, ) < Rk; n, ) < k) Rk; n k, ) and by iterating the alication of 4), for m = k we obtain the bounds: 5) ) k n k+ ) < Rk; n, ) < k n k A stronger lower bound for Rk; n, ), obtained also by alication of Theorem 2.6, is the following theorem: Theorem 2.7 Th.2 and Cor.2 in [9]). For every k n hk, ), where hk, ) = and lk, ) =, the following inequalities holds: k q n k hk,)+ 6) Rk; n, ) k) n k+ lk,)hk,) ) 7) k ) n k+ lk,)hk,) ) k ) n k+ Referring to the limiting behavior of L n, Gordon, Schilling and Waterman roved in [6] some results regarding the distribution of the longest k-interruted head run L k n), and they also managed to generalize the results obtained by Boyd in [2]. They analyzed the head runs by using geometric random variables as follows: let Y 0 ), Y 0 ),... be an i.i.d. sequence of geometric random variables with arameter q =, so that P Y 0 n) = m) = q m, and Sm) = m + m j= Y 0j) for m and Sm) = 0 otherwise. The values of Sm) reresent the locations of T s in the sequence X, X 2,... of heads and tails, where X n = {n Sj), j>0}. It is easy to see

9 that Y 0 m) reresents the length of the m-th comleted ure head run, which at the Sm) toss ends by a T. If we write M 0 n) = max m n Y 0 m), and we take Nn) to reresent the number of tails in the firs n tosses, then the length of the longest head run is defined by L n = max {M 0 Nn)), n SNn))}. We have to observe that we used only the subscrit 0 because the general treatment will be given in the last art. Their main results for k = 0) are: Theorem 2.8 Th. in [6]). Let W ) have an ) extreme standard distribution, i.e. PW x) = e e x, and δ 0 n) = ln ln. Then, uniformly in t, as n. PL n log qn) t) P W ) + δ 0 qn) δ 0 qn) t 0 ln which shows that L n is well aroximated by M 0 Nn)). Versions of next theorem can be found also in Boyd [2] and in Guibas and Odlyzko [7]: Theorem 2.9 Th.2 in [6]). Let θ = 8) ln π2 ). Then EL n) log qn) + γ ) ln 2 < θ 2 2πe θ e θ ) 2 + o) 9 and 9) V arl n ) 6 ln π 2 )) < θ)θ 2 2πe θ e θ ) 3 + o). It can be seen that the results obtained in Theorem 2.8 and in Theorem 2.9 are in close relation with the equations ) and 2) obtained by Boyd in [2]. We will finish this section by characterizing the limiting behavior for the robability that the longest run is formed by heads successes). We notate this robability by V n ) and we notice that V n ) = PL n > L n ), where L n reresents the length of the longest failure tail) run in the n trials. Theorem 2.0 Th.5 in [9]). The robability V n ) has the following limit values { 0, if 0 < 2 20) lim V n ) = n, if <. 2

10 0 The Theorem 2.0 can be interreted as follows: let A and B be two layers that toss a biassed coin that give a good outcome for the first layer, i.e. for examle heads with robability, and a good outcome for the second layer as tails with robability q = ; and consider that after n tosses wins the layer who has scored the longest sequence of favorable outcomes. Now if the number of tosses increase indefinitely then the layer which has the greater robability of getting favorable outcomes will win for sure. 3. Extensions and further results As we announced in the revious sections there are many other tyes of run henomena related to the length of the longest run. One of these henomena that we will describe in what follows is the longest k-interruted head run L k n). Erdös and Révész studied this roblem and showed in [5] see also Révész [2]) some limiting roerties which are generalizations of the Theorem 2. and Theorem 2.3) for the longest k-interruted head run, but only in the case of fair coins. Other generalizations can be found also in Guibas and Odlyzko [7] and in Gordon, Schilling and Waterman in [6] for examle generalizations for the Theorem 2.8 and Theorem 2.9). We will give these results below, but let s do some notations first. Consider that we are in the situation described before the Theorem 2.8 and consider now the case of the head runs interruted by kt s. Denote the summed lengths of k + ure head run starting with run m and ending with run m + k, lus the kt s that searate the k + comonent head runs, by Y k m) = Sm + k) + Sm )), and note that the random variable Y k m) k = k j=0 Y 0m + j) has negative binomial distribution with arameters k +, q). Write M k n) = max m n Y k m), when n > 0 and 0 otherwise and define the length of the longest k-interruted head run in the first n tosses by L k n) = max {M k Nn) k), n SNn) k)} where, as before, ) Nn) reresents the number of tails in the firs n tosses. Letting λ = ln and ) q µ k n) = log n) + k log log n) + k log log k!) we have: Theorem 3. generalization of Theorem 2.8). Let W have an extreme standard distribution, i.e. PW x) = e e x, and δ k n) = µ k n) µ k n). Then, uniformly in t, ) W PL k n) µ k qn) t) P λ + δ kqn) δ k qn) t 0 as n.

11 Theorem 3.2 generalization of Theorem 2.9). Let θ = π2. Then λ 2) EL kn)) µ k qn) + γ λ θ < 2) 2 2πe θ e θ ) 2 + o) and 22) π 2 V arl kn)) 6λ + ) < θ)θ πe θ e θ ) 3 + o). Regarding the almost sure behavior of the longest k-interruted head run, Guibas and Odlyzko in [7] by means of generating functions in theorems 3 and 3) and Gordon, Schilling and Waterman in [6], resented a comlete characterization of the function which are touched infinitely often by usually long or short longest head runs. We will remark that the following theorems are generalizations of the results obtained by Erdös and Révész in [5] Theorem 2.): Theorem 3.3 Th.3 in [6]). Let α n a non-decreasing sequence of integers. Then { 0, if α k 23) PL k n) α n i.o.) = n αn <, if αn k αn =. A similar theorem, but for PL k n) < α n i.o.) is given in [6], but require more care and is based on the construction of a sequence of countinous random variables, Y k ˆj), with a given distribution function and such that the max j n Y k ˆ j)+k µ k qn) W λ and Y k j) = Y k ˆ j) + k. We will finish this section by giving a result due to Mori [0] that is in close relation with the results described by Gordon, Schilling and Waterman [6]: Theorem 3.4 Cor.5. in [0]). The following relation is valid almost sure: n t+ 24) lim n lnn) i { } = e k! q ) q k z dz. L i n) log i) k log log i)<t i= t

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