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In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero. $$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \mathop {\lim }\limits_{\Delta t \to 0} \sum\nolimits_{m = 1}^{M - 1} {{\bf{\tilde q}}\left( {\frac{{{\tau _m} + {\tau _{m + 1}}}}{2}} \right)\Delta t} $$ How we can reach from integral definition to summation definition ?

${\bf{\tilde q}}(t)$ is white Gaussian noise process.

EDIT:

We can see Wiener process as the limit of a random walk as we let the sample time go to zero. Now the question is,

How we can relate the Wiener process to random walk using the definition above?

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sci9
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    All integrals are limits of a sum. – Simon Boge Brant May 27 '19 at 09:55
  • The statement is just a definition. The right hand side "explains" what is meant by the notation on the left hand side. – Simon Boge Brant May 27 '19 at 10:02
  • Thank you for comment. Cloud you tell me what Capital $M$ stands for? – sci9 May 27 '19 at 10:20
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    To define the integral over the interval [0, t], you make a partition of the interval into M -1 pieces, defined by the $M$ points $\tau_1, \dots \tau_M$ along the interval. The length of one of them is here denoted $\Delta t$, the distance from $\tau_m$ to $\tau_{m+1}$. The integral is then (here) defined as limit of the sum of the function evaluated on the midpoint of each of these segments, when we let the size of each segment get smaller and smaller (and thus also let the number of points M inrease). – Simon Boge Brant May 27 '19 at 10:51
  • +1 Thank you for clarifying things. Could you add your comment as an answer? – sci9 May 27 '19 at 10:57
  • @SimonBogeBrant Could you please take a look at my question https://stats.stackexchange.com/q/410267/137591? – sci9 May 27 '19 at 10:58
  • sci9, although you have been treated to a sketch of the Riemann integral of real-valued functions in these comments, it's a long leap to defining this integral of stochastic processes. That leap covers questions of what kind of limit this is and whether it exists. Your question refers to an "integral definition," so please tell us what definition of this stochastic integral you are working with (there are several and they don't always give the same results!). – whuber May 27 '19 at 11:56
  • @whuber please see edits – sci9 May 27 '19 at 13:29
  • Thank you. This might come down to what exactly you mean by "random walk," because (at least according to some definitions), the sum on the right hand side is precisely a "limit of a random walk as we let the sample time [interval] go to zero." How does this differ from your understanding of a random walk? – whuber May 27 '19 at 13:52
  • @whuber Thank you for clarifying things. My question is exactly why and how the right hand can be seen as a random walk? – sci9 May 27 '19 at 14:19
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    +1 I consulted your reference. It's sloppy and nonsensical, in no small part due to all the typographical errors. It does't even define the $\tau_m$ or explain how the limit can possibly make any sense (there is, explicitly, a different limiting process for each $t,$ thereby raising significant questions concerning how $w(t)$ and $w(s)$ might be related for $s\ne t$). Thus, what you need is a full account of how Brownian motion can be achieved as a limit of scaled, symmetric random walks. Consider using a different resource for learning this material! – whuber May 27 '19 at 15:05
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    @whuber Thank you very much for suggestions. I searched your keywords and the result is: "the BM can be obtained as a limit of scaled discrete-time symmetric random walks". So my question is how we can express this sentense mathematically? How we can express mathematically $\bf{w}(t)$ process as random walks? How and when we can convert the integral of this stochastic process to summation? How we can mathematically express "limit of scaled, symmetric random walks" and relate it to integral of BM? – sci9 May 27 '19 at 15:37

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