Pertimbangkan jalan acak 1 dimensi pada bilangan bulat dengan keadaan awal x ∈ Z :
di mana kenaikan adalah IID sehingga P { ξ i = 1 } = P { ξ i = - 1 } = 1 .
Seseorang dapat membuktikan bahwa (1)
di mana subscript menunjukkan posisi awal.
Biarkan menjadi waktu bagian pertama yang negara + 1 . Dengan kata lain, τ : = τ ( 1 ) : = min { n ≥ 0 : S n = 1 } . Orang juga dapat membuktikan bahwa (2)
Kedua bukti tersebut dapat ditemukan di http://galton.uchicago.edu/~lalley/Courses/312/RW.pdf . Melalui membaca artikel, saya mengerti kedua bukti.
Namun, pertanyaan saya adalah apa arti "akhirnya" dalam pernyataan pertama dan juga secara umum. Jika sesuatu terjadi "pada akhirnya", itu tidak harus terjadi dalam waktu yang terbatas, bukan? Jika demikian, apa sebenarnya perbedaan antara sesuatu yang tidak terjadi dan sesuatu yang tidak terjadi "pada akhirnya"? Pernyataan (1) dan (2) dalam beberapa hal bertentangan dengan saya. Apakah ada contoh lain seperti ini?
EDIT
Hanya ingin menambahkan motivasi untuk pertanyaan, yaitu, contoh langsung dari sesuatu yang terjadi "akhirnya", tetapi dengan waktu tunggu yang terbatas diharapkan.
Oleh karena itu kita tahu bahwa walker akan "akhirnya" bergerak ke kiri, dan waktu tunggu yang diharapkan sebelum melakukannya (yaitu, bergerak kiri) adalah .
Melihat sesuatu yang terjadi "akhirnya" tetapi dengan "waktu tunggu" yang tak terbatas diharapkan cukup imajinasi saya. Bagian kedua dari respons @ whuber adalah contoh hebat lainnya.
Jawaban:
How would you demonstrate an event "eventually happens"? You would conduct a thought experiment with a hypothetical opponent. Your opponent may challenge you with any positive numberp . If you can find an n (which most likely depends on p ) for which the chance of the event happening by time n is at least 1−p , then you win.
In the example, "Sn " is misleading notation because you use it both to refer to one state of a random walk as well as to the entire random walk itself. Let's take care to recognize the distinction. "Reaches 1 eventually" is meant to refer to a subset S of the set of all random walks Ω . Each walk S∈Ω has infinitely many steps. The value of S at time n is Sn . "S reaches 1 by time n " refers to the subset of Ω of walks that have reached the state 1 by time n . Rigorously, it is the set
In your response to the imaginary opponent, you are exhibiting someΩ1,n with the property that
Becausen is arbitrary, you have available all elements of the set
(Recall thatS∈⋃∞n=1Ω1,n if and only if there exists a finite n for which S∈Ω1,n , so there aren't any infinite numbers involved in this union.)
Your ability to win the game shows this union has a probability exceeding all values of the form1−p , no matter how small p>0 may be. Consequently, that probability is at least 1 --and therefore equals 1 . You will have demonstrated, then, that
One simple way to appreciate the distinction between "happening eventually" and having an infinite expected first passage time is to contemplate a simpler situation. Forn any natural number, let ω(n) be the sequence
in whichn zeros are followed by an endless string of ones. In other words, these are the walks that stay at the origin and at some (finite) time step over to the point 1 , then stay there forever.
LetΩ be the set of all these ω(n),n=0,1,2,… with the discrete sigma algebra. Assign a probability measure via
This was designed to make the chance of jumping to1 by the time n equal to 1−1/(n+1) , which obviously approaches arbitrarily closely to 1 . You will win the game. The jump eventually happens and when it does, it will be at some finite time. However, the expected time when it happens is the sum of the survival function (which gives the chances of not having jumped at time n ),
which diverges. That is because a relatively large probability is given to waiting a long time before jumping.
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in the description ofThat something happens eventually means that there is some point in time at which it happens, but there is a connotation that one is not referring to any particular specified time before which it happens. If you say something will happen within three weeks, that is a stronger statement than that it will happen eventually. That it will happen eventually does not specify a time, such as "three weeks" or "thirty-billion years" or "one minute".
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