ESG Transition matrix (Spring 2022)
[Notice] Journey to the academic researcher This is the story of how I became the insightful researcher.
The transition matrix gives the probability of transitioning from one ESG rating state to any other state. That transition matrix would, of course, just reflect probabilities for the single period observed here. Assuming that single period sample can be considered representative, what are the steady state probabilities of ending up with each rating? That is, what is the steady state vector of this markov process?
import numpy as np
import pandas as pd
df=pd.read_csv('msci_timeseries_20210901.csv')
rating = ['CCC','B','BB','BBB','A','AA','AAA']
cross = pd.crosstab(df['IVA_COMPANY_RATING'], df['IVA_PREVIOUS_RATING'], normalize = 'columns')[['CCC','B','BB','BBB','A','AA','AAA']]
cross = cross.reindex(rating, axis = "rows")
cross
| IVA_PREVIOUS_RATING | CCC | B | BB | BBB | A | AA | AAA |
|---|---|---|---|---|---|---|---|
| IVA_COMPANY_RATING | |||||||
| CCC | 0.68 | 0.045267 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| B | 0.28 | 0.563786 | 0.064356 | 0.004082 | 0.000000 | 0.000000 | 0.000000 |
| BB | 0.04 | 0.374486 | 0.514851 | 0.095918 | 0.004237 | 0.000000 | 0.000000 |
| BBB | 0.00 | 0.016461 | 0.391089 | 0.591837 | 0.116525 | 0.011050 | 0.000000 |
| A | 0.00 | 0.000000 | 0.029703 | 0.289796 | 0.629237 | 0.102210 | 0.016807 |
| AA | 0.00 | 0.000000 | 0.000000 | 0.018367 | 0.241525 | 0.825967 | 0.067227 |
| AAA | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.008475 | 0.060773 | 0.915966 |
np_cross = cross.to_numpy()
pd.set_option('display.max_columns', 20)
def matrix_power(matrix, power):
if power == 0:
return np.identity(len(matrix))
elif power == 1:
return matrix
else:
return np.dot(matrix, matrix_power(matrix, power-1))
matrix_power(np_cross, 200).round(3)
array([[0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001],
[0.005, 0.005, 0.005, 0.005, 0.005, 0.005, 0.005],
[0.022, 0.022, 0.022, 0.022, 0.022, 0.022, 0.022],
[0.087, 0.087, 0.087, 0.087, 0.087, 0.087, 0.087],
[0.191, 0.191, 0.191, 0.191, 0.191, 0.191, 0.191],
[0.392, 0.392, 0.392, 0.392, 0.392, 0.392, 0.392],
[0.303, 0.303, 0.303, 0.303, 0.303, 0.303, 0.303]])
matrix_power(np_cross, 200).sum(axis=0)
array([1., 1., 1., 1., 1., 1., 1.])
for i in range(1,100,1):
print(f'n Step Transition Matrix at the nth power {i}\n', matrix_power(np_cross, i).round(3),'\n')
n Step Transition Matrix at the nth power 1
[[0.68 0.045 0. 0. 0. 0. 0. ]
[0.28 0.564 0.064 0.004 0. 0. 0. ]
[0.04 0.374 0.515 0.096 0.004 0. 0. ]
[0. 0.016 0.391 0.592 0.117 0.011 0. ]
[0. 0. 0.03 0.29 0.629 0.102 0.017]
[0. 0. 0. 0.018 0.242 0.826 0.067]
[0. 0. 0. 0. 0.008 0.061 0.916]]
n Step Transition Matrix at the nth power 2
[[0.475 0.056 0.003 0. 0. 0. 0. ]
[0.351 0.355 0.071 0.011 0.001 0. 0. ]
[0.153 0.407 0.327 0.109 0.016 0.001 0. ]
[0.02 0.165 0.437 0.422 0.147 0.028 0.003]
[0.001 0.016 0.147 0.359 0.455 0.153 0.033]
[0. 0. 0.014 0.096 0.354 0.711 0.121]
[0. 0. 0. 0.004 0.028 0.107 0.843]]
n Step Transition Matrix at the nth power 3
[[0.339 0.054 0.005 0.001 0. 0. 0. ]
[0.341 0.243 0.064 0.015 0.002 0. 0. ]
[0.231 0.361 0.238 0.102 0.025 0.004 0. ]
[0.078 0.265 0.405 0.335 0.15 0.043 0.007]
[0.011 0.07 0.231 0.361 0.366 0.179 0.048]
[0.001 0.007 0.055 0.174 0.407 0.632 0.165]
[0. 0. 0.002 0.012 0.051 0.142 0.78 ]]
n Step Transition Matrix at the nth power 4
[[0.246 0.048 0.006 0.001 0. 0. 0. ]
[0.302 0.176 0.054 0.017 0.003 0.001 0. ]
[0.268 0.304 0.186 0.092 0.029 0.007 0.001]
[0.143 0.31 0.361 0.283 0.145 0.055 0.012]
[0.036 0.132 0.275 0.345 0.317 0.192 0.062]
[0.005 0.028 0.109 0.238 0.431 0.576 0.2 ]
[0. 0.001 0.007 0.025 0.074 0.17 0.725]]
n Step Transition Matrix at the nth power 5
[[0.181 0.041 0.007 0.001 0. 0. 0. ]
[0.257 0.134 0.046 0.017 0.004 0.001 0. ]
[0.275 0.255 0.152 0.082 0.032 0.01 0.002]
[0.199 0.321 0.321 0.246 0.139 0.064 0.017]
[0.073 0.185 0.295 0.327 0.288 0.199 0.075]
[0.015 0.061 0.164 0.287 0.44 0.534 0.229]
[0.001 0.004 0.016 0.04 0.097 0.193 0.677]]
n Step Transition Matrix at the nth power 6
[[0.135 0.034 0.007 0.002 0. 0. 0. ]
[0.214 0.104 0.039 0.016 0.005 0.001 0. ]
[0.264 0.215 0.128 0.074 0.033 0.012 0.003]
[0.238 0.314 0.286 0.219 0.133 0.071 0.022]
[0.113 0.223 0.3 0.309 0.269 0.202 0.087]
[0.034 0.101 0.213 0.323 0.442 0.503 0.253]
[0.002 0.009 0.027 0.057 0.118 0.211 0.634]]
n Step Transition Matrix at the nth power 7
[[0.101 0.028 0.006 0.002 0. 0. 0. ]
[0.176 0.083 0.033 0.015 0.006 0.002 0. ]
[0.245 0.182 0.109 0.066 0.033 0.015 0.004]
[0.261 0.299 0.257 0.199 0.128 0.076 0.027]
[0.151 0.248 0.298 0.294 0.256 0.203 0.098]
[0.06 0.144 0.256 0.349 0.44 0.48 0.273]
[0.005 0.016 0.04 0.074 0.137 0.225 0.597]]
n Step Transition Matrix at the nth power 8
[[0.077 0.023 0.006 0.002 0.001 0. 0. ]
[0.145 0.068 0.029 0.014 0.006 0.002 0.001]
[0.222 0.156 0.095 0.06 0.032 0.016 0.005]
[0.272 0.28 0.233 0.182 0.123 0.079 0.032]
[0.184 0.263 0.292 0.282 0.246 0.203 0.107]
[0.091 0.185 0.291 0.368 0.437 0.462 0.29 ]
[0.01 0.026 0.055 0.092 0.155 0.237 0.564]]
n Step Transition Matrix at the nth power 9
[[0.059 0.018 0.005 0.002 0.001 0. 0. ]
[0.118 0.056 0.025 0.013 0.006 0.003 0.001]
[0.198 0.134 0.083 0.055 0.032 0.018 0.007]
[0.272 0.26 0.213 0.168 0.119 0.082 0.037]
[0.211 0.271 0.285 0.271 0.239 0.202 0.116]
[0.125 0.223 0.319 0.382 0.433 0.448 0.304]
[0.016 0.037 0.07 0.109 0.17 0.247 0.536]]
n Step Transition Matrix at the nth power 10
[[0.045 0.015 0.005 0.002 0.001 0. 0. ]
[0.097 0.046 0.022 0.012 0.006 0.003 0.001]
[0.176 0.117 0.074 0.051 0.031 0.019 0.008]
[0.267 0.242 0.196 0.157 0.116 0.084 0.041]
[0.231 0.273 0.277 0.262 0.233 0.202 0.124]
[0.161 0.257 0.341 0.391 0.429 0.437 0.316]
[0.024 0.05 0.086 0.125 0.184 0.255 0.51 ]]
n Step Transition Matrix at the nth power 11
[[0.035 0.012 0.004 0.002 0.001 0. 0. ]
[0.08 0.039 0.019 0.011 0.006 0.003 0.001]
[0.155 0.102 0.066 0.047 0.03 0.02 0.009]
[0.257 0.224 0.181 0.148 0.113 0.086 0.045]
[0.244 0.273 0.27 0.254 0.228 0.201 0.131]
[0.195 0.286 0.358 0.398 0.425 0.428 0.326]
[0.034 0.063 0.102 0.141 0.197 0.262 0.487]]
n Step Transition Matrix at the nth power 12
[[0.028 0.01 0.004 0.002 0.001 0. 0. ]
[0.066 0.033 0.017 0.011 0.006 0.004 0.002]
[0.137 0.09 0.06 0.044 0.03 0.021 0.01 ]
[0.245 0.208 0.169 0.14 0.11 0.087 0.049]
[0.253 0.27 0.262 0.247 0.224 0.2 0.137]
[0.227 0.311 0.371 0.402 0.422 0.421 0.334]
[0.045 0.078 0.117 0.155 0.208 0.268 0.467]]
n Step Transition Matrix at the nth power 13
[[0.022 0.008 0.003 0.002 0.001 0. 0. ]
[0.055 0.028 0.015 0.01 0.006 0.004 0.002]
[0.121 0.08 0.055 0.041 0.029 0.021 0.011]
[0.231 0.194 0.158 0.133 0.108 0.087 0.053]
[0.258 0.266 0.256 0.241 0.22 0.199 0.143]
[0.256 0.331 0.38 0.405 0.418 0.416 0.342]
[0.057 0.092 0.132 0.169 0.218 0.272 0.45 ]]
n Step Transition Matrix at the nth power 14
[[0.017 0.007 0.003 0.002 0.001 0. 0. ]
[0.046 0.024 0.014 0.009 0.006 0.004 0.002]
[0.107 0.072 0.05 0.039 0.029 0.022 0.012]
[0.218 0.181 0.149 0.128 0.105 0.088 0.056]
[0.26 0.261 0.249 0.236 0.217 0.198 0.148]
[0.282 0.347 0.388 0.406 0.415 0.411 0.348]
[0.07 0.107 0.147 0.181 0.227 0.277 0.434]]
n Step Transition Matrix at the nth power 15
[[0.014 0.006 0.003 0.002 0.001 0. 0. ]
[0.038 0.021 0.012 0.009 0.006 0.004 0.002]
[0.095 0.065 0.046 0.037 0.028 0.022 0.013]
[0.205 0.17 0.142 0.123 0.104 0.088 0.059]
[0.26 0.256 0.244 0.231 0.214 0.198 0.153]
[0.305 0.36 0.393 0.407 0.413 0.408 0.353]
[0.083 0.121 0.16 0.193 0.235 0.28 0.42 ]]
n Step Transition Matrix at the nth power 16
[[0.011 0.005 0.002 0.001 0.001 0.001 0. ]
[0.032 0.018 0.011 0.008 0.006 0.004 0.002]
[0.084 0.059 0.043 0.035 0.027 0.022 0.014]
[0.193 0.16 0.135 0.119 0.102 0.088 0.062]
[0.258 0.251 0.239 0.227 0.212 0.197 0.157]
[0.324 0.371 0.397 0.407 0.41 0.405 0.358]
[0.097 0.135 0.172 0.203 0.242 0.283 0.407]]
n Step Transition Matrix at the nth power 17
[[0.009 0.004 0.002 0.001 0.001 0.001 0. ]
[0.028 0.016 0.01 0.008 0.006 0.004 0.003]
[0.076 0.054 0.041 0.033 0.027 0.022 0.015]
[0.181 0.152 0.129 0.115 0.1 0.088 0.064]
[0.256 0.247 0.234 0.223 0.21 0.196 0.16 ]
[0.34 0.379 0.4 0.407 0.408 0.403 0.362]
[0.111 0.149 0.184 0.213 0.249 0.285 0.396]]
n Step Transition Matrix at the nth power 18
[[0.007 0.004 0.002 0.001 0.001 0.001 0. ]
[0.024 0.014 0.01 0.007 0.006 0.004 0.003]
[0.068 0.049 0.038 0.032 0.027 0.022 0.015]
[0.171 0.144 0.124 0.112 0.099 0.088 0.066]
[0.252 0.242 0.23 0.22 0.208 0.196 0.164]
[0.353 0.386 0.401 0.406 0.406 0.401 0.366]
[0.124 0.161 0.195 0.221 0.254 0.288 0.386]]
n Step Transition Matrix at the nth power 19
[[0.006 0.003 0.002 0.001 0.001 0.001 0. ]
[0.021 0.013 0.009 0.007 0.005 0.004 0.003]
[0.062 0.046 0.036 0.031 0.026 0.023 0.016]
[0.161 0.137 0.12 0.109 0.098 0.088 0.068]
[0.249 0.238 0.226 0.217 0.206 0.195 0.166]
[0.364 0.39 0.402 0.406 0.405 0.399 0.369]
[0.137 0.173 0.205 0.229 0.259 0.289 0.377]]
n Step Transition Matrix at the nth power 20
[[0.005 0.003 0.002 0.001 0.001 0.001 0. ]
[0.018 0.012 0.008 0.007 0.005 0.004 0.003]
[0.056 0.043 0.034 0.03 0.026 0.023 0.017]
[0.153 0.131 0.116 0.106 0.097 0.088 0.07 ]
[0.245 0.233 0.223 0.214 0.204 0.195 0.169]
[0.373 0.394 0.403 0.405 0.403 0.398 0.371]
[0.15 0.184 0.214 0.237 0.264 0.291 0.369]]
n Step Transition Matrix at the nth power 21
[[0.004 0.002 0.001 0.001 0.001 0.001 0. ]
[0.016 0.011 0.008 0.006 0.005 0.005 0.003]
[0.052 0.04 0.033 0.029 0.025 0.023 0.017]
[0.145 0.126 0.113 0.104 0.096 0.088 0.072]
[0.241 0.23 0.22 0.212 0.203 0.194 0.171]
[0.38 0.397 0.403 0.404 0.402 0.397 0.374]
[0.162 0.195 0.222 0.243 0.268 0.292 0.362]]
n Step Transition Matrix at the nth power 22
[[0.004 0.002 0.001 0.001 0.001 0.001 0. ]
[0.014 0.01 0.007 0.006 0.005 0.005 0.003]
[0.048 0.038 0.032 0.028 0.025 0.023 0.018]
[0.139 0.122 0.11 0.102 0.095 0.088 0.073]
[0.237 0.226 0.217 0.21 0.202 0.194 0.173]
[0.386 0.398 0.403 0.403 0.401 0.396 0.376]
[0.174 0.204 0.23 0.249 0.272 0.294 0.356]]
n Step Transition Matrix at the nth power 23
[[0.003 0.002 0.001 0.001 0.001 0.001 0. ]
[0.013 0.009 0.007 0.006 0.005 0.005 0.003]
[0.044 0.036 0.031 0.028 0.025 0.023 0.018]
[0.133 0.118 0.107 0.101 0.094 0.088 0.075]
[0.233 0.223 0.214 0.208 0.201 0.194 0.175]
[0.39 0.4 0.403 0.402 0.4 0.396 0.378]
[0.185 0.213 0.237 0.254 0.275 0.295 0.35 ]]
n Step Transition Matrix at the nth power 24
[[0.003 0.002 0.001 0.001 0.001 0.001 0. ]
[0.011 0.008 0.007 0.006 0.005 0.005 0.003]
[0.041 0.034 0.03 0.027 0.025 0.023 0.019]
[0.128 0.114 0.105 0.099 0.093 0.088 0.076]
[0.229 0.22 0.212 0.206 0.2 0.194 0.177]
[0.393 0.4 0.402 0.401 0.399 0.395 0.379]
[0.195 0.222 0.243 0.259 0.278 0.296 0.345]]
n Step Transition Matrix at the nth power 25
[[0.002 0.002 0.001 0.001 0.001 0.001 0. ]
[0.01 0.008 0.006 0.006 0.005 0.005 0.004]
[0.039 0.033 0.029 0.027 0.024 0.023 0.019]
[0.123 0.111 0.103 0.098 0.093 0.088 0.077]
[0.226 0.217 0.21 0.205 0.199 0.193 0.178]
[0.396 0.401 0.402 0.401 0.398 0.394 0.381]
[0.204 0.229 0.249 0.264 0.28 0.296 0.341]]
n Step Transition Matrix at the nth power 26
[[0.002 0.001 0.001 0.001 0.001 0.001 0. ]
[0.009 0.007 0.006 0.006 0.005 0.005 0.004]
[0.037 0.031 0.028 0.026 0.024 0.023 0.019]
[0.119 0.108 0.101 0.097 0.092 0.088 0.078]
[0.223 0.215 0.208 0.203 0.198 0.193 0.18 ]
[0.397 0.401 0.401 0.4 0.397 0.394 0.382]
[0.213 0.236 0.254 0.268 0.283 0.297 0.337]]
n Step Transition Matrix at the nth power 27
[[0.002 0.001 0.001 0.001 0.001 0.001 0. ]
[0.009 0.007 0.006 0.005 0.005 0.005 0.004]
[0.035 0.03 0.027 0.026 0.024 0.023 0.02 ]
[0.115 0.106 0.1 0.096 0.091 0.088 0.079]
[0.22 0.212 0.206 0.202 0.197 0.193 0.181]
[0.398 0.401 0.401 0.399 0.397 0.394 0.383]
[0.221 0.242 0.259 0.271 0.285 0.298 0.333]]
n Step Transition Matrix at the nth power 28
[[0.002 0.001 0.001 0.001 0.001 0.001 0.001]
[0.008 0.007 0.006 0.005 0.005 0.005 0.004]
[0.033 0.029 0.027 0.025 0.024 0.023 0.02 ]
[0.112 0.104 0.098 0.095 0.091 0.088 0.08 ]
[0.217 0.21 0.205 0.201 0.197 0.193 0.182]
[0.399 0.401 0.4 0.398 0.396 0.393 0.384]
[0.229 0.248 0.264 0.274 0.287 0.298 0.33 ]]
n Step Transition Matrix at the nth power 29
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.008 0.006 0.006 0.005 0.005 0.005 0.004]
[0.032 0.028 0.026 0.025 0.024 0.023 0.02 ]
[0.109 0.102 0.097 0.094 0.091 0.088 0.08 ]
[0.215 0.208 0.203 0.2 0.196 0.193 0.183]
[0.4 0.4 0.399 0.398 0.396 0.393 0.385]
[0.236 0.254 0.267 0.277 0.288 0.299 0.327]]
n Step Transition Matrix at the nth power 30
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.007 0.006 0.006 0.005 0.005 0.005 0.004]
[0.031 0.028 0.026 0.025 0.024 0.023 0.02 ]
[0.107 0.1 0.096 0.093 0.09 0.088 0.081]
[0.212 0.207 0.202 0.199 0.196 0.192 0.184]
[0.4 0.4 0.399 0.397 0.395 0.393 0.385]
[0.242 0.258 0.271 0.28 0.29 0.299 0.325]]
n Step Transition Matrix at the nth power 31
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.007 0.006 0.005 0.005 0.005 0.005 0.004]
[0.03 0.027 0.025 0.024 0.023 0.023 0.021]
[0.104 0.099 0.095 0.092 0.09 0.088 0.082]
[0.21 0.205 0.201 0.198 0.195 0.192 0.185]
[0.4 0.399 0.398 0.397 0.395 0.393 0.386]
[0.248 0.263 0.274 0.282 0.291 0.3 0.322]]
n Step Transition Matrix at the nth power 32
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.007 0.006 0.005 0.005 0.005 0.005 0.004]
[0.029 0.027 0.025 0.024 0.023 0.023 0.021]
[0.102 0.098 0.094 0.092 0.09 0.087 0.082]
[0.209 0.204 0.2 0.198 0.195 0.192 0.185]
[0.4 0.399 0.398 0.396 0.395 0.393 0.387]
[0.253 0.267 0.277 0.284 0.292 0.3 0.32 ]]
n Step Transition Matrix at the nth power 33
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.006 0.006 0.005 0.005 0.005 0.005 0.004]
[0.028 0.026 0.025 0.024 0.023 0.023 0.021]
[0.101 0.096 0.093 0.091 0.089 0.087 0.083]
[0.207 0.202 0.199 0.197 0.194 0.192 0.186]
[0.399 0.398 0.397 0.396 0.394 0.392 0.387]
[0.258 0.27 0.28 0.286 0.293 0.3 0.318]]
n Step Transition Matrix at the nth power 34
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.006 0.006 0.005 0.005 0.005 0.005 0.004]
[0.027 0.026 0.025 0.024 0.023 0.023 0.021]
[0.099 0.095 0.093 0.091 0.089 0.087 0.083]
[0.205 0.201 0.198 0.196 0.194 0.192 0.187]
[0.399 0.398 0.397 0.395 0.394 0.392 0.388]
[0.262 0.273 0.282 0.288 0.294 0.3 0.317]]
n Step Transition Matrix at the nth power 35
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.006 0.005 0.005 0.005 0.005 0.005 0.004]
[0.027 0.025 0.024 0.024 0.023 0.023 0.021]
[0.098 0.094 0.092 0.09 0.089 0.087 0.084]
[0.204 0.2 0.198 0.196 0.194 0.192 0.187]
[0.399 0.397 0.396 0.395 0.394 0.392 0.388]
[0.266 0.276 0.284 0.289 0.295 0.301 0.315]]
n Step Transition Matrix at the nth power 36
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.006 0.005 0.005 0.005 0.005 0.005 0.004]
[0.026 0.025 0.024 0.024 0.023 0.023 0.021]
[0.097 0.094 0.091 0.09 0.089 0.087 0.084]
[0.203 0.199 0.197 0.195 0.194 0.192 0.188]
[0.398 0.397 0.396 0.395 0.393 0.392 0.388]
[0.27 0.279 0.286 0.291 0.296 0.301 0.314]]
n Step Transition Matrix at the nth power 37
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.006 0.005 0.005 0.005 0.005 0.005 0.004]
[0.026 0.025 0.024 0.023 0.023 0.023 0.021]
[0.096 0.093 0.091 0.09 0.088 0.087 0.084]
[0.202 0.199 0.196 0.195 0.193 0.192 0.188]
[0.398 0.396 0.395 0.394 0.393 0.392 0.389]
[0.273 0.281 0.288 0.292 0.297 0.301 0.313]]
n Step Transition Matrix at the nth power 38
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.025 0.024 0.024 0.023 0.023 0.023 0.022]
[0.095 0.092 0.091 0.089 0.088 0.087 0.084]
[0.201 0.198 0.196 0.195 0.193 0.192 0.188]
[0.397 0.396 0.395 0.394 0.393 0.392 0.389]
[0.276 0.283 0.289 0.293 0.297 0.301 0.312]]
n Step Transition Matrix at the nth power 39
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.025 0.024 0.024 0.023 0.023 0.022 0.022]
[0.094 0.092 0.09 0.089 0.088 0.087 0.085]
[0.2 0.197 0.195 0.194 0.193 0.192 0.189]
[0.397 0.396 0.395 0.394 0.393 0.392 0.389]
[0.278 0.285 0.29 0.294 0.298 0.301 0.311]]
n Step Transition Matrix at the nth power 40
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.025 0.024 0.023 0.023 0.023 0.022 0.022]
[0.093 0.091 0.09 0.089 0.088 0.087 0.085]
[0.199 0.197 0.195 0.194 0.193 0.192 0.189]
[0.396 0.395 0.394 0.394 0.393 0.392 0.39 ]
[0.281 0.287 0.292 0.295 0.298 0.301 0.31 ]]
n Step Transition Matrix at the nth power 41
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.025 0.024 0.023 0.023 0.023 0.022 0.022]
[0.092 0.091 0.09 0.089 0.088 0.087 0.085]
[0.198 0.196 0.195 0.194 0.193 0.192 0.189]
[0.396 0.395 0.394 0.393 0.393 0.392 0.39 ]
[0.283 0.289 0.293 0.296 0.299 0.302 0.309]]
n Step Transition Matrix at the nth power 42
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.024 0.024 0.023 0.023 0.023 0.022 0.022]
[0.092 0.09 0.089 0.089 0.088 0.087 0.085]
[0.197 0.196 0.194 0.193 0.193 0.192 0.189]
[0.396 0.395 0.394 0.393 0.393 0.392 0.39 ]
[0.285 0.29 0.294 0.296 0.299 0.302 0.308]]
n Step Transition Matrix at the nth power 43
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.024 0.024 0.023 0.023 0.023 0.022 0.022]
[0.091 0.09 0.089 0.088 0.088 0.087 0.085]
[0.197 0.195 0.194 0.193 0.192 0.192 0.19 ]
[0.395 0.394 0.394 0.393 0.392 0.392 0.39 ]
[0.287 0.291 0.295 0.297 0.299 0.302 0.308]]
n Step Transition Matrix at the nth power 44
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.024 0.023 0.023 0.023 0.023 0.022 0.022]
[0.091 0.09 0.089 0.088 0.088 0.087 0.086]
[0.196 0.195 0.194 0.193 0.192 0.192 0.19 ]
[0.395 0.394 0.393 0.393 0.392 0.392 0.39 ]
[0.288 0.292 0.295 0.298 0.3 0.302 0.307]]
n Step Transition Matrix at the nth power 45
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.024 0.023 0.023 0.023 0.023 0.022 0.022]
[0.09 0.089 0.089 0.088 0.088 0.087 0.086]
[0.196 0.194 0.194 0.193 0.192 0.192 0.19 ]
[0.395 0.394 0.393 0.393 0.392 0.392 0.39 ]
[0.29 0.294 0.296 0.298 0.3 0.302 0.307]]
n Step Transition Matrix at the nth power 46
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.024 0.023 0.023 0.023 0.023 0.022 0.022]
[0.09 0.089 0.088 0.088 0.087 0.087 0.086]
[0.195 0.194 0.193 0.193 0.192 0.192 0.19 ]
[0.394 0.394 0.393 0.393 0.392 0.392 0.391]
[0.291 0.294 0.297 0.298 0.3 0.302 0.306]]
n Step Transition Matrix at the nth power 47
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.09 0.089 0.088 0.088 0.087 0.087 0.086]
[0.195 0.194 0.193 0.193 0.192 0.192 0.19 ]
[0.394 0.393 0.393 0.393 0.392 0.392 0.391]
[0.292 0.295 0.297 0.299 0.301 0.302 0.306]]
n Step Transition Matrix at the nth power 48
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.089 0.089 0.088 0.088 0.087 0.087 0.086]
[0.195 0.194 0.193 0.192 0.192 0.192 0.19 ]
[0.394 0.393 0.393 0.393 0.392 0.392 0.391]
[0.293 0.296 0.298 0.299 0.301 0.302 0.306]]
n Step Transition Matrix at the nth power 49
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.089 0.088 0.088 0.088 0.087 0.087 0.086]
[0.194 0.193 0.193 0.192 0.192 0.192 0.19 ]
[0.394 0.393 0.393 0.392 0.392 0.392 0.391]
[0.294 0.297 0.298 0.3 0.301 0.302 0.305]]
n Step Transition Matrix at the nth power 50
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.089 0.088 0.088 0.088 0.087 0.087 0.086]
[0.194 0.193 0.193 0.192 0.192 0.192 0.191]
[0.394 0.393 0.393 0.392 0.392 0.392 0.391]
[0.295 0.297 0.299 0.3 0.301 0.302 0.305]]
n Step Transition Matrix at the nth power 51
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.089 0.088 0.088 0.088 0.087 0.087 0.086]
[0.194 0.193 0.192 0.192 0.192 0.191 0.191]
[0.393 0.393 0.393 0.392 0.392 0.392 0.391]
[0.296 0.298 0.299 0.3 0.301 0.302 0.305]]
n Step Transition Matrix at the nth power 52
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.023 0.022 0.022]
[0.089 0.088 0.088 0.087 0.087 0.087 0.086]
[0.193 0.193 0.192 0.192 0.192 0.191 0.191]
[0.393 0.393 0.392 0.392 0.392 0.392 0.391]
[0.297 0.298 0.3 0.3 0.301 0.302 0.305]]
n Step Transition Matrix at the nth power 53
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.088 0.088 0.087 0.087 0.087 0.086]
[0.193 0.193 0.192 0.192 0.192 0.191 0.191]
[0.393 0.393 0.392 0.392 0.392 0.392 0.391]
[0.297 0.299 0.3 0.301 0.301 0.302 0.304]]
n Step Transition Matrix at the nth power 54
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.088 0.088 0.087 0.087 0.087 0.086]
[0.193 0.193 0.192 0.192 0.192 0.191 0.191]
[0.393 0.393 0.392 0.392 0.392 0.392 0.391]
[0.298 0.299 0.3 0.301 0.302 0.302 0.304]]
n Step Transition Matrix at the nth power 55
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.088 0.087 0.087 0.087 0.087 0.087]
[0.193 0.192 0.192 0.192 0.192 0.191 0.191]
[0.393 0.392 0.392 0.392 0.392 0.392 0.391]
[0.298 0.299 0.3 0.301 0.302 0.302 0.304]]
n Step Transition Matrix at the nth power 56
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.088 0.087 0.087 0.087 0.087 0.087]
[0.193 0.192 0.192 0.192 0.192 0.191 0.191]
[0.393 0.392 0.392 0.392 0.392 0.392 0.391]
[0.299 0.3 0.301 0.301 0.302 0.302 0.304]]
n Step Transition Matrix at the nth power 57
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.004]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.088 0.087 0.087 0.087 0.087 0.087]
[0.193 0.192 0.192 0.192 0.192 0.191 0.191]
[0.393 0.392 0.392 0.392 0.392 0.392 0.391]
[0.299 0.3 0.301 0.301 0.302 0.302 0.304]]
n Step Transition Matrix at the nth power 58
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.023 0.023 0.022 0.022 0.022]
[0.088 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.192 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.299 0.3 0.301 0.301 0.302 0.302 0.304]]
n Step Transition Matrix at the nth power 59
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.023 0.022 0.022 0.022 0.022]
[0.088 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.192 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.3 0.301 0.301 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 60
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.023 0.022 0.022 0.022 0.022]
[0.088 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.192 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.3 0.301 0.301 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 61
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.023 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.192 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.3 0.301 0.301 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 62
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.192 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.301 0.301 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 63
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.301 0.301 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 64
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.023 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.391]
[0.301 0.301 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 65
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.301 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 66
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.301 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 67
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.301 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 68
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.023 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.192 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.301 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 69
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 70
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 71
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.192 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.302 0.303]]
n Step Transition Matrix at the nth power 72
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 73
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 74
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.192 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 75
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 76
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 77
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 78
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.192 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 79
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 80
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 81
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 82
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 83
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 84
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.302 0.303 0.303]]
n Step Transition Matrix at the nth power 85
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 86
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 87
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 88
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 89
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 90
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.302 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 91
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 92
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 93
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.302 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 94
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.303 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 95
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.303 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 96
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.302 0.303 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 97
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.303 0.303 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 98
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.303 0.303 0.303 0.303 0.303 0.303]]
n Step Transition Matrix at the nth power 99
[[0.001 0.001 0.001 0.001 0.001 0.001 0.001]
[0.005 0.005 0.005 0.005 0.005 0.005 0.005]
[0.022 0.022 0.022 0.022 0.022 0.022 0.022]
[0.087 0.087 0.087 0.087 0.087 0.087 0.087]
[0.191 0.191 0.191 0.191 0.191 0.191 0.191]
[0.392 0.392 0.392 0.392 0.392 0.392 0.392]
[0.302 0.303 0.303 0.303 0.303 0.303 0.303]]