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]]

![enter image description here][1]


  [1]: https://i.sstatic.net/M0bUY.png