Correct step back on age diagonal #2

src/vonFoersterHazards.ipynb

@@ -130,19 +130,19 @@
     "\\begin{array}{rcl}\n",
     "    \\mathbb{C}ov\\left[ N^{\\left(a, \\Delta t\\right)}_i , N^{\\left(a, \\Delta t\\right)}_j \\right]\n",
     "    & = &\n",
-    "    \\mathbb{E}\\left[\\mathbb{C}ov\\left[ N^{\\left(a,\\Delta t\\right)}_i , N^{\\left(a,\\Delta t\\right)}_j \\parallel \\vec{N}^{\\left(a,0\\right)} \\right]\\right]\n",
+    "    \\mathbb{E}\\left[\\mathbb{C}ov\\left[ N^{\\left(a,\\Delta t\\right)}_i , N^{\\left(a,\\Delta t\\right)}_j \\parallel \\vec{N}^{\\left(a - \\Delta t,0\\right)} \\right]\\right]\n",
     "    +\n",
-    "    \\mathbb{C}ov\\left[ \\mathbb{E}\\left[ N^{\\left(a,\\Delta t\\right)}_i \\parallel \\vec{N}^{\\left(a,0\\right)} \\right] , \\mathbb{E}\\left[ N^{\\left(a,\\Delta t\\right)}_j \\parallel \\vec{N}^{\\left(a,0\\right)} \\right] \\right]\\\\\\\\\n",
+    "    \\mathbb{C}ov\\left[ \\mathbb{E}\\left[ N^{\\left(a,\\Delta t\\right)}_i \\parallel \\vec{N}^{\\left(a - \\Delta t,0\\right)} \\right] , \\mathbb{E}\\left[ N^{\\left(a,\\Delta t\\right)}_j \\parallel \\vec{N}^{\\left(a - \\Delta t,0\\right)} \\right] \\right]\\\\\\\\\n",
     "    & = &\n",
     "    \\left\\{\n",
     "    \\begin{array}{ll}\n",
-    "        \\displaystyle \\sum_k \\mathbb{E}\\left[N^{\\left(a,0\\right)}_k\\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow i}\\right] \\left(1-\\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow i}\\right]\\right)\n",
+    "        \\displaystyle \\sum_k \\mathbb{E}\\left[N^{\\left(a - \\Delta t,0\\right)}_k\\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow i}\\right] \\left(1-\\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow i}\\right]\\right)\n",
     "        +\n",
-    "        \\sum_{k,l} \\mathbb{C}ov \\left[ N^{\\left(a,0\\right)}_k, N^{\\left(a,0\\right)}_l \\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{l \\rightarrow i}\\right]\n",
+    "        \\sum_{k,l} \\mathbb{C}ov \\left[ N^{\\left(a - \\Delta t,0\\right)}_k, N^{\\left(a - \\Delta t,0\\right)}_l \\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{l \\rightarrow i}\\right]\n",
     "        & i=j \\\\\\\\\n",
-    "        \\displaystyle \\sum_k \\mathbb{E}\\left[N^{\\left(a,0\\right)}_k\\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow j}\\right]\n",
+    "        \\displaystyle \\sum_k \\mathbb{E}\\left[N^{\\left(a - \\Delta t,0\\right)}_k\\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow j}\\right]\n",
     "        +\n",
-    "        \\sum_{k,l} \\mathbb{C}ov \\left[ N^{\\left(a,0\\right)}_k, N^{\\left(a,0\\right)}_l \\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a\\right)}_{l \\rightarrow j}\\right]\n",
+    "        \\sum_{k,l} \\mathbb{C}ov \\left[ N^{\\left(a - \\Delta t,0\\right)}_k, N^{\\left(a - \\Delta t,0\\right)}_l \\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{k \\rightarrow i}\\right] \\mathbb{P}\\left[T^{\\left(a - \\Delta t\\right)}_{l \\rightarrow j}\\right]\n",
     "        & \\text{otherwise}\n",
     "    \\end{array}\n",
     "    \\right.\n",