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A straightforward way would be using recursion and memoization. An example:

n = 5;
c = RandomReal[NormalDistribution[], n]/100;

Clear[x]
Array[(x[#] = RandomReal[NormalDistribution[]]) &, n]; (* Initial conditions *)
x[t_Integer] /; t > n := x[t] = c.Table[x[i], {i, t - 1, t - n, -1}] +
     RandomReal[NormalDistribution[]]

ListLinePlot[x /@ Range[300]]