はぐれ競馬論文(Stray Horse Racing paper)
(2025年3月22日公開、2025年04月01日01:32:56第4回の改訂)
競馬ホーム(Horse racing Home)・ホーム・English Home

ここでは、結局審査を通らなかった、雑誌掲載をあきらめた論文を紹介しています。
It publishes an article that have not been peer-reviewed.



本ページでの公開の背景というか理由 Reason
2025/3/2雑誌Aに提出Submitted Journal A
2025/3/8雑誌Aの編集者から、雑誌の目的とずれているという指摘あり。雑誌Bへの提出を推薦。Editor of Journal A said that this paper doesn't fit the journal's purpose. He recommended Journal B.
2025/3/11雑誌Bに短縮版を提出Submitted short version to Journal B
2025/3/16雑誌Bの編集者から書式などがガイドラインに合っていないので一旦却下する。ガイドラインを読み直して再提出してくださいとの連絡。Editor of Journal B refused my paper because the format doesn't follow guidelines.
2025/3/22ページ数を減らしたり、書式を合わせるのが手間に感じるし、新しいネタの準備に時間を使いたいので、再提出を断念。このページでの公開を決意Reducing number of pages is a hassle. It's also a hassle to format. I have some new topics and want to write them. I decided to make it public rather than publishing it in a journal.



概要 Abstract
In this paper, I try to generalize the method of doubling (the rule of doubles) such that rewards can be obtained with lower risk. Since it is said that applying doubling generally leads to bankruptcy, I introduce an expansion method so that the risk is reduced. That is, instead of doubling the bet after a loss, we wait to increase the bet until the uncollected amount reaches a predetermined value. I enter to compete in a single win of horse racing at a virtual racecourse by betting the same rank or the same horse number rather than predicting. Further, I propose some improved models, skip low odds races, skip all races after a win until the next day, and start betting with a low-odds first-rank horse. Next, I evaluate models for how much of an input amount is needed to be risked. Then, I show the results generated by 100,000 races. It is discovered that the input amount may be increased 10,000 times, which may dissuade many people from using this method. Finally, I propose some ideas for future study, including betting methods for multiple horses, and for one-horse multiple-lines methods. While risk cannot be completely eliminated, we can observe upward trending except for some deep valleys in profit graphs. Horse racing is a special type of gambling whose past data is publicly available. This includes data such as hit rate, recovery rate, and average refunds by rank or horse number. I believe that this study can become a foothold to develop a winning method in the future.


Some Challenges for the Improvement of Doubling at a Virtual Racecourse


Section 1 Introduction
In this paper, I attempt to make a profit with low risk by using an extension of the method of doubling. We are able to find data in Japan where the recovery rate is 80% or over in single-win betting for predictions of first place. So, my goal is to increase the recovery rate to over 100% with low risk by betting the same rank or same horse number rather than using predictions.
Doubling is well-known as a dubious winning method. The strategy is simple: Keep doubling the bet after losing until a win is achieved. If we have enough funds and the return, or payout, is over 200%, then a profit over 100% is guaranteed. Turner (1998) compares doubling to constant betting by using computer simulations. Applying this to horse racing is widely discussed. We can find many websites and movies about applying doubling to horse racing through search engines by using the keywords “Martingale methods” and “horse racing”. A simple application of this method is as follows: Bet on a horse whose odds is around 2, then double the bet after losing. Some websites propose alterations or modifications. Umameshi.com proposes some alterations of doubling that are focused on odds, for example. Similar to the standard doubling method, these alterations also carry the risk of insufficient funds.
Here, I discuss an expansion of this method which suppresses the input amount (the wager) to our tolerable range and reduces the risk of bankruptcy. The method is as follows: We predetermine the value that we will wait to reach in order to increase the bet. We compute the input amount using this value and the uncollected amount (the temporary accumulated loss). After winning such that uncollected amount is zero, we set the input amount to the initial amount. By repeating this, we expect to see improvement of the recovery rate with a small amount of funds. As a risk evaluation index, we use the maximum input amount in 10,000 races.
In Section 2, I explain the simulation data used in the following sections. Section 2 is not related to the methods discussed in the following sections and can be skipped if there is no interest in the data creation process.
In Section 3, I define constants and variables, and I formulate the standard model. In Section 4, I show examples of two strategies. One strategy is betting on horses of the same rank, and the other is betting on the same horse number. Predictions are not used. In Japan, horse numbers are positioned from the inside. I show how the profit changes in both good and bad situations. Further, I discuss the effects of changing parameter values.
In Sections 5 through 7, I propose some improved methods. I design several models that aim to reduce the maximum input amount to 100,000 yen or less over 10,000 races. There are two reasons why I set the upper limit to 100,000 yen. One is based on my experience of using doubling with an initial bet of 100 yen. After a long losing streak, it becomes mentally difficult to continue to use this strategy. Confidence is lost in one’s own betting, which is not necessarily due to their remaining account balance. The other reason is the possibility of making the odds lowered and the returns reduced due to one’s own invested bets. In Japan, aside from graded races, it seems like that there are not many large bets over 100,000 yen. (I don’t have accurate information about this.)
In Section 5, using fixed parameters and a predetermined rank or horse number, some improvements are proposed such that the input amount is lowered by a reduction of betting frequency. That is, under certain conditions we skip some races, skip low-odds races when an uncollected amount remains, skip all remaining races after a win such that the uncollected amount becomes zero, and start betting at the predetermined low-odds, most popular horse whose probability of winning is high.
In Section 6, for reduction of risk in the case that the uncollected amount becomes large, I propose increasing the parameters so that we can store a higher uncollected amount.
Further, in Section 7, I propose a method for betting on the lower-ranked horses which can store larger uncollected amounts as a workaround for the case where the uncollected amount becomes large.
In Section 8, I show the results of 100,000 races using the parameters and models constructed in previous sections. In actuality, I constructed the models such that the input amount does not exceed 100,000 yen. I confirmed whether the input amount exceeded 100,000 yen in 100,000 races in order to look ahead to actual operation in practical use. There were a few cases where the input amount exceeded 100,000 yen. Furthermore, in 100,000 races, there were also some cases where the input amount grew to 10,000 times the initial amount. Due to the fear of doubling, we cannot say that it is safe to use the model in 10,000 races, and we cannot determine how much preparation would be necessary for actual operation in practical use. This is what I want to emphasize most in this paper.
In Sections 9 and 10, I describe some ideas for future study. In Section 9, I try to recover from a large uncollected amount. In Section 10, I introduce a multiple-line model. For the case of betting on multiple horses in the same race, I show how to bet on one horse divided into multiple lines to reduce the increase of the input amount. In the multiple-line model, we can reduce risk by clearing the uncollected amount when the total profit is updated. Moreover, it is effective to use the multiple-line model in the case where the uncollected amount becomes large in the one-line model.
Finally, in Section 11, I conclude this paper. In addition, although it may be superfluous, I write about my dreams and delusions. While there is a risk that the input amount grows to over 100,000 yen, we can confirm that graphs show that profits are generally increasing. There are some techniques which lead to reducing this risk. I hope this paper will serve as a stepping stone for the development of a reliable method of winning.
Gerolamo Cardano said that the biggest advantage in gambling is not to gamble at all: “Il vantaggio piu grande del gioco d'azzardo sta nel non giocare affatto.” I aim to challenge this quotation.



Section 2 to 10, please download pdf file.
Some Challenges for the Improvement of Doubling at a Virtual Racecourse

Section 11 End talk
11.1 Conclusion
In this paper, I proposed a relaxation method for the rule of doubles in order to make a profit with low risk at a virtual racecourse. In these models, rather than using predictions, we continued to bet on the same rank or the same horse number. I designed several models by using the data gathered from 10,000 races. I confirmed the amount of risk present in the data through 100,000 races, with the forward-looking goal of practical use of the models in future races. We observed that, while a model may appear to be safe over a short timeframe, up to 10,000 times the input amount may be needed.
In Section 2, I explained the method of constructing the virtual horse racing data. A large amount of racing data is necessary for discussions regarding betting strategy. Although the accuracy may not be high enough, I believe it is a meaningful example.
In Section 3, I defined parameters and variables. I also introduced the standard model. The input amount is calculated by the unclaimed winnings and a predetermined coefficient. By this configuration, the rate of increase of the input amount can be slowed down.
In Section 4, I introduced good periods and bad periods in the standard model. I also explained the importance of setting coefficient b.
In Section 5, I proposed the reduction of betting frequency. By this operation, the maximum input amount, Amax, has improved from being an astronomically high value to a more realistic value.
In Section 6, I discussed increasing coefficient b when the uncollected amount becomes large.
Additionally, in Section 7, I proposed the strategy of lowering the horse rank as a workaround when the uncollected amount becomes even larger.
In Section 8, looking towards practical application, I showed the results of 100,000 races by applying a model that was found to be good in 10,000 races with initial parameters. This resulted in some cases where the input amount grows to 10,000 times its initial value. This is the most important conclusion of this paper.
Sections 9 and 10 were included as subjects for future work. In Section 9, I explained that we can not recover from a large loss by simple methods.
In Section 10, I introduced the one-line multiple-horses model, the one-horse multiple-lines model, and the multiple-lines multiple-sets model. These models seem attractive because (1) the multiple-lines model allows for dividing the uncollected amount, (2) the multiple-horses model allows for raising the hit rate, and (3) the one-horse multiple-lines model allows for slowing down the rate of increase of the uncollected amount. The multiple-lines model may be more effective if used when the uncollected amount becomes large. Further, when we use a large number of lines, the betting counts for each line are reduced, so we might be able to prevent sudden declines as seen in the previous sections. We may be able to develop a periodic system similar to those seen with massive outbreaks of periodic cicadas.
In consideration of actual practical application, there are many problems. First of all, I used artificial racing data. I don’t know how much of a difference there is compared to actual data. I gave examples with the assumption that I knew which methods have a recovery rate of 80%. But I don’t know how to bet in order to obtain an 80% recovery rate. Even if I found some data where the recovery rate is 80% within a certain time period, the next time period might not even reach 50%. In Japanese newspapers, tipsters use marks such as ◎, 〇, ▲, and △, but these marks are often intended for use in key horse betting methods such as wheel and box. Some tipsters’ use of the mark ◎ might not even fulfill an 80% recovery rate. With wheel and box betting, the amount of the bet made at one time tends to be high, so these are not suitable for doubling if we do not have enough funds to cover the bet.
There is also a problem with the amount of data. I used data from 100,000 races for verifying the safety of the models. However, in some cases, many races were skipped. For example, with the Nn strategy of Type III in Section 5.2, 88% of the races were skipped, as shown in Fig. 6(c). I would like to claim that I have generated a sufficient amount of test data. However, if someone pointed out that I only used 12% in some cases, I would have to acknowledge that. In addition, I don’t know how many tests would be sufficient. Skipping races to reduce betting counts is the most important strategy in Section 5. This allows us to steer clear of the most easily avoidable losing streaks.
It’s not just a financial issue, but also a psychological burden. As for myself, I put these methods into practice before writing this paper, and I felt a lot of mental stress. For the popular horses, we must set parameter b to a low enough value to obtain winnings. However, even if the hit rate is relatively high, the recovery rate might change. Even if the recovery rate is over 80% within a certain time period, it may be different in the next time period. I tried methods based on using the most popular horses and the prediction symbols in newspapers, but I often felt discouraged by sudden increases in the betting amount. Sufficient research and preparation is necessary. For the less popular horses, we can set parameter b to a large value, but we must wait a long time for a dark horse long shot to win. I once tried waiting through 48 races over two weekends for the third horse from the bottom (the third lowest-ranked horse) to win. I know that statistically it should occur at a 1% or 2% hit rate, but I don’t know when it will occur. While the payoff is large, the psychological stress is unbearable and there is no estimate for how much time it will take. When I bet on the third horse from the bottom over a long period, it’s not a problem if a popular horse comes in. That loss is endurable. But if the fourth horse from the bottom comes in, then I feel regret. If the second horse from the bottom comes in, it’s more painful, and I regret choosing the third horse instead of the second horse. Repeat that experience over and over. If I had endured the trial and my horse eventually came in, I would have earned a big win. This was the case for just 48 races. I doubt whether waiting over a longer period is possible.

11.2 Future study
In Section 5, I focused on the reduction of the number of bets. I believe that this is one of the more effective methods. In televised programs, some tipsters will skip a number of races, and we can observe voters skipping some races as well. They must know from experience that skipping is an effective strategy. However, finding more effective methods to reduce the number of bets is a topic for future study.
In Section 10, I introduced the multiple-lines model. I believe that the one-line model is better than the multiple-lines model. However, the multiple-lines model is effective when the uncollected amount is large. Going forward, we might be able to create a periodic system.
In Section 5.4, I used constant betting data and proposed starting from low-odds races. Even if using the doubling method is theoretically safe, the psychological stress is inevitable, so we would like to avoid using doubling as much as possible. Ideally, we want to develop a winning strategy which does not require using the doubling method. We should only use doubling as a last resort when we cannot obtain winnings by constant betting. Developing a more effective combination method is a topic for future study.
In Section 9, I explained that recovering from a large loss is difficult. In the first place, we don’t know the amount of funds we would need to obtain winnings. If there is a debt, we can pay it off in installments. However, in doubling, there is a risk that the divided uncollected amount may become large again. For now, it is important to develop methods such that uncollected amount remains under a predetermined value.
I believe that the methods proposed in this paper have properties similar to the iterative methods for computing f(x)=0 as with the Newton method. Hundreds of papers have been published in the field of numerical calculation. Studies of convergence conditions might contribute some improvement to the doubling method. Actually, while I was changing the value of b in Section 6, I was recalling the change of the slope of the Newton method.
In the field of stochastic processes, similar discussions often take place where convergence conditions are deeply discussed. As a result, I hope that we can find conditions in the future that allow us to obtain winnings from lower betting amounts. The rule of doubles was referred to as the Martingale method somewhat abruptly in the field of horse racing. However, this is a term that was originally used in the context of conditional probability. (See Nishiyama, 2021).
The figures for profit changes remind me of Fourier analysis, especially those of the multiple-lines model. We can interpret hit rates periodically and the multiple-lines model as a superposition of their functions.
Study of the Traveling Salesman Problem and Steiner’s Problem might also contribute to future study. Heuristic algorithms are developed and productivity is largely improved. These ideas might also contribute to the doubling method.
In this paper, I constructed the data such that the average of returns is made close to actual Japanese races. However, I still think there is a big difference between these data and those of real races. The results based on these data should be taken with a grain of salt. Higher quality data will be needed for more accurate analysis.
Further, I only bet on single-win ticket types?that is, win bets. Different methods will be needed for ticket types such as place, quinella place, bracket quinella, quinella, exacta, trio, and trifecta. For other types of public gambling, such as boat racing and bicycle races, I think these methods can be applied by using the same parameters. However, from the viewpoint of acquiring historical race data, single-win tickets in horse racing are the easiest.
These methods may also be applicable to investments such as stocks, exchange rates, and futures trading. I don’t yet know how to set the parameters, though there might be some hints in the large amount of investment information that is updated daily.
Since I show examples generated by using artificial data, valid parameter values might differ. I avoided using real-world data because I didn’t know how effective the historical data was. I cannot bear any responsibility if impatient readers go bankrupt by applying a proposed model. I only believe that there must exist some methods and parameters that can increase winnings in horse racing. There are many factors at play. Basically, the strong horses win. Depending on the course, there are advantages and disadvantages for each horse number and each gate number. We can refer to statistical data for historical recovery rates and hit rates, and we can also consider the distance between the gate and the first corner. There is also a characteristic similar to the property of memorylessness. That is, the probability of extending the losing streak of the most popular horse is independent of the current length of its losing streak. As these properties are further and more deeply analyzed, the accuracy of forecasting for future investment should be improved.

11.3 Considering the emergence of winning methods
It is interesting to consider the world after the development of successful methods of winning.
I don’t wish for young people to quit their jobs and immerse themselves in horse racing. However, it would be nice to make life easier. Gambling has led to a lot of social problems including terrible tragedies, addiction, and dependence. I hope that successful methods of winning can help to prevent these unfortunate situations.
I think that when people stop excessively saving and start spending their money, it leads to enriching our country. First of all, 10% of sales from the purchase of horse racing tickets is paid to the national treasury in Japan. Not to mention, workers in the horse racing industry can be saved. Furthermore, in recent times, horses and horse racing have become factors in international trade. Stallions are traded actively, and many Japanese horses participate in overseas races. I hope that a Japanese horse will win the Prix de l’Arc de Triomphe title someday. I expect that the income from horse racing may also be able to provide support for the independence of poor countries.
On the contrary, will the value of the currency of countries who have horse racing decline against those of countries who do not? Or, will the value of money disappear? Will the value of food ever exceed the value of money? In ancient times, humans didn’t use money. Animals other than humans also don’t use money. When currency was developed, people thought it was convenient. We have inherited the tools of our predecessors. Meanwhile, the disadvantages of money have been neglected. Since food spoils, it is better to distribute it rather than having to store large quantities. Money doesn’t rot, so it can be kept in banks for future use instead of being distributed to the poor. Can we call this convenient? Osamu Dazai claimed that if people didn’t know that Mt. Fuji was famous, they wouldn’t be impressed. Since everyone says it is beautiful, people are impressed by the mountain. (Having said that, I do love this mountain.) Similarly, just as our parents used money for convenience, we also use it without question. If aliens were observing this situation, what would they say? Why do they use money? Is money convenient? Is this despite the fact that many people are killed not by food or by love, but by money?

11.4 What horse racing teaches us
First of all, horse racing teaches us many variations of how to use money. I can enjoy a stable life with no debt by renting an inexpensive apartment. Not knowing a thing about horse racing, I could not take out a housing loan due to an indistinct feeling of anxiety. What would I do if something happened to me? Without overcoming this anxiety, I still lived in a lower-class apartment. This way of thinking changed after experiencing the ups and downs of income and expenses as a result of purchasing horse racing tickets. If I knew about horse racing and how to bet on horses to make money, I could take out a housing loan and acquire my dream home. Buying a house is a high-risk undertaking. However, I have learned how to compete. Because I’m no longer young, I cannot take out a loan. From this introspection, I have determined that if these methods become well-known, I would buy a home. This is my gamble. If I bought a horse, I could be the owner of a Prix de l’Arc de Triomphe prize horse. Even before winning a single bet, the gambler’s fantasy is endless.
Further, horse racing tells us that our forecasting is pretty sloppy even after many attempts. It is said that the most popular horse loses over 60% of the time, and the recovery rate is about 80%. Although I don’t intend to reject democracy, I think that the rights of election winners are just too expansive. Election winners simply gained popularity after completing training and paddock work. It is easy to know how many times the most popular horse wins by the fanfare of the trumpets. Yet how can we tell how many election winners end up leaving good results? Not to mention, voters must research candidates more seriously than they do horse races. Many gamblers check the historical race data for horse racing. Similarly, we need to reflect on our voting behavior and the performance of our candidates. Candidates who lose elections did not lose the race just because they did not receive enough votes. It would be beneficial to accumulate their accomplishments for the next election.
In horse racing, it is possible to recognize when profits are diminishing. We can adjust our betting methods, change our strategy, or eventually stop betting, as some do. If we seriously want to obtain profits, we can employ the PDCA (Plan-Do-Check-Act) cycle as we plot how the profit changes race-by-race. It is possible to discover some effective strategies in that cycle, such as the reduction of the number of betting tickets and betting amounts. On the other hand, as an example from daily life, even if we keep a household account ledger, it may not be possible to improve profits because there are too many factors involved. We may not realize that we are being held back by things that aren’t actually important. Personally, I think this way of thinking has enabled me to throw away things that were not necessary to hold on to, and this improved my personal finances. Cardano said that not betting is the best strategy for gamblers. Similarly, not buying and not owning may be the best strategy for life.
Please forgive me for continuing to ramble on. I would also like to mention the use of historical data. In Section 8, I showed that the doubling models that show good results in data0 often lead to catastrophic results in 100,000 races. Can this be considered a special circumstance unique to doubling? We frequently discuss safety by using historical data. However, in the fields of climate and earthquakes, for example, we often experience damage that exceeds the limits of historical data. This suggests that it may be a mistake to argue for safety based solely on historical data. We must consider alternative approaches to demonstrating safety. In the case of horse racing, this is not a problem, because the only person who is harmed is the gambler themselves.
There are many people who don’t participate in horse racing because they think they will lose money. I think it’s true that you’ll lose money. But if you try it and think that you’re throwing away money, you’ll discover something. I recommend that you decide on a time limit and a maximum amount and try it out.

11.5 Background of this paper
Despite my poor English skills, there are three reasons why I wrote this paper in English.
The first reason is that I want people all over the world to know about these methods.
The second reason is to prevent the bankruptcy of impatient readers who come away with misunderstandings. In English, it would be difficult to extract an excerpt and implement it for Japanese people. On the other hand, for non-Japanese people, it is difficult to implement the proposed models because of the different currencies and circumstances around horse racing.
The third reason is tax. From the viewpoint of investment, taxes cannot be ignored. In Japan, the purchase cost of a losing betting ticket usually cannot be treated as an expense. According to the guidelines of the National Tax Agency, the conditions that allow for losing betting tickets to be treated as expenses are extremely strict. As long as this system is in place, if the profit from winning betting tickets exceeds 500,000 yen in one year (excluding losing betting tickets), then we must pay taxes calculated on this amount combined with other income. Fig. 30 shows some examples of cases where there are 10,000 races in a year. The upper lines are the profit changes for computing taxes, and the lower lines are the actual profits including the cost of losing betting tickets. Even if the lower lines fall below zero, if the upper lines surpass 500,000, then we must pay additional taxes. Since total taxes are dependent on all other income, they cannot be shown easily in figures. However, as shown in Section 8, I can claim that after one year, profits based on doubling might be negative, as we might have purchased more betting tickets that lost. Under these conditions, models based on doubling cannot be used for investment.


Fig. 30. Profits subject to tax calculation.

I learned about horse racing by chance. In January 2021, while I was visiting Obihiro (a remote city in Hokkaido) on a business trip, I stopped by the city’s race course, which is a tourist attraction in the area. There I found a horse named Trumpet. Since playing trumpet is my hobby, I bet on the horse. Fortunately, with what must be beginner’s luck, the horse came in first place. I became addicted to horse racing after that. Although I was just a beginner, I researched gambling and horse racing and found doubling to be a suspicious winning method. I tried doubling while making improvements to the method. I could make a few hundred yen, but at one point I incurred a loss of over 100,000 yen. I looked further into the historical data and started over. Once again, I made a few hundred yen, and I felt like the probability of a big loss had gone down a bit. Just then, I again incurred a loss of over 100,000 yen. At that moment, I realized how risky doubling can be. Using models based on about 1,000 past races leads to bankruptcy. I decided to create artificial race data based on simulations of tens of thousands of races before implementing a model. I resolved to make sure there were no pitfalls in the model. Even when good results were seen (e.g., Amax=5,900 with N4 of Type III in Table 33), I still didn’t feel like putting the model into practice. The amount of data is small, and it could be just luck. Eventually, a deep valley will appear. I still had doubts. Yet, I continued going to the off-track betting shop every weekend. I purchased tickets for my favorite jockeys and for tickets related to my birthday for about 3,000 yen, limited to graded races. While doing this, I was looking for materials to build better models.
In gambling, there are good times and bad times. The bad times are long lasting, and the good times don’t last long. Even if there was something that could lead to a method of winning, many scientists, including myself, haven’t been paying enough attention. Like astrology, we do not consider it to be a good subject for research. Cardano’s famous quotation also has an adverse effect. The forms of gambling that he targeted are mainly card games, dice, and roulette. It is said that the birthplace of modern horse racing was 16th-century England, and that it was introduced to Italy in the 18th century. It is not clear whether Cardano knew of the types of gambling such as with horse racing. However, times have evolved, and more horse racing data is now available. While I constructed the racing data in Section 2 using the assumption that the odds and the winner are decided by a roulette wheel, I often think that it is interesting that we assume the winner is decided by roulette wheels in actual horse races. We can construct roulette wheels from historical data including hit rates by rank, hit rates by horse number, win rates of jockeys, win rates by odds, and so on. If we observe the races from these viewpoints, we might be able to enjoy them more. I believe we have a chance to create solid methods for winning.

It is regrettable that there are some cases in 100,000 races where we have to prepare enough funds to cover 10,000 times the input amount. Even if we seem to be able to develop good methods through 10,000 races, there is a large risk when applied in 100,000 races. However, excluding some deep valleys, we can observe upward trending. I believe that there is no reason to give up.
Cardano’s quotation has not yet been proven correct. According to a legend about horse racing, there are some people who use reliable methods of winning and who make large profits. I don’t know the specific details or whether they have anything to do with doubling. Data analysis and model considerations related to horse racing are interesting, and sometimes I get so engrossed in my research that I can’t sleep. At times like this, I recall Italian opera. Cardano’s quotation will be overcome in the not-so-distant future. Then, with my favorite musicians, I will sing Nessun Dorma out loud like Luciano Pavarotti. “Vincero!”


References
Box G. EP, Muller M. E. (1958). A note on the generation of random normal deviates. Ann. Math. Stat., 29, 610-611,.
Nishiyama Y., (2021). Martingale Methods in Statistics, Chapman & Hall.
Turner N. E., (1998). Doubling vs. Constant Bets as Strategies for Gambling, J. of Gambling Studies, 14, 413-429.
Umameshi.com, (Last update 2021). https://www.umameshi.com/info/002.html [In Japanese]
Yoshimura J. et al., (2009). Selection For Prime-Number Intervals In A Numerical Model Of Periodical Cicada Evolution, Evolution, 63(1), 288-294, https://doi.org/10.1111/j.1558-5646.2008.00545.x




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