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##### MSD within the AR model

The relation between the VACF and the MSD reads

| (4.84)

By discretizing the above equation one obtains

| (4.85)

Using

|  |  | (4.86)

into Eq. 4.85, gives

| (4.87)

Making use of the one-side z-transform (equivalent to the Laplace transform for discrete functions), we obtain

| (4.88)

where

| (4.89)

Introducing Eq. 4.89 into Eq. 4.88 yields

| (4.90)

and its inverse z-transform reads

| (4.91)

Using the expression of the non-normalized obtained in the framework of the AR model

| (4.92)

one finds the expression of MSD within the same AR model

|  |  | (4.93)
|  |  | (4.94)

| (4.95)

which allows one to compute the MSD within the AR model from the poles and the coefficients of the non-normalized VACF.

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